12.3: Examples of Static Equilibrium
- Identify and analyze static equilibrium situations
- Set up a free-body diagram for an extended object in static equilibrium
- Set up and solve static equilibrium conditions for objects in equilibrium in various physical situations
All examples in this chapter are planar problems. Accordingly, we use equilibrium conditions in the component form of Equation 12.2.9 to Equation 12.2.11 . We introduced a problem-solving strategy in Example 12.1 to illustrate the physical meaning of the equilibrium conditions. Now we generalize this strategy in a list of steps to follow when solving static equilibrium problems for extended rigid bodies. We proceed in five practical steps.
- Identify the object to be analyzed. For some systems in equilibrium, it may be necessary to consider more than one object. Identify all forces acting on the object. Identify the questions you need to answer. Identify the information given in the problem. In realistic problems, some key information may be implicit in the situation rather than provided explicitly.
- Set up a free-body diagram for the object. (a) Choose the xy-reference frame for the problem. Draw a free-body diagram for the object, including only the forces that act on it. When suitable, represent the forces in terms of their components in the chosen reference frame. As you do this for each force, cross out the original force so that you do not erroneously include the same force twice in equations. Label all forces—you will need this for correct computations of net forces in the x- and y-directions. For an unknown force, the direction must be assigned arbitrarily; think of it as a ‘working direction’ or ‘suspected direction.’ The correct direction is determined by the sign that you obtain in the final solution. A plus sign (+) means that the working direction is the actual direction. A minus sign (−) means that the actual direction is opposite to the assumed working direction. (b) Choose the location of the rotation axis; in other words, choose the pivot point with respect to which you will compute torques of acting forces. On the free-body diagram, indicate the location of the pivot and the lever arms of acting forces—you will need this for correct computations of torques. In the selection of the pivot, keep in mind that the pivot can be placed anywhere you wish, but the guiding principle is that the best choice will simplify as much as possible the calculation of the net torque along the rotation axis.
- Set up the equations of equilibrium for the object. (a) Use the free-body diagram to write a correct equilibrium condition Equation 12.2.9 for force components in the x-direction. (b) Use the free-body diagram to write a correct equilibrium condition Equation 12.2.13 for force components in the y-direction. (c) Use the free-body diagram to write a correct equilibrium condition Equation 12.2.11 for torques along the axis of rotation. Use Equation 12.2.12 to evaluate torque magnitudes and senses.
- Simplify and solve the system of equations for equilibrium to obtain unknown quantities. At this point, your work involves algebra only. Keep in mind that the number of equations must be the same as the number of unknowns. If the number of unknowns is larger than the number of equations, the problem cannot be solved.
- Evaluate the expressions for the unknown quantities that you obtained in your solution. Your final answers should have correct numerical values and correct physical units. If they do not, then use the previous steps to track back a mistake to its origin and correct it. Also, you may independently check for your numerical answers by shifting the pivot to a different location and solving the problem again, which is what we did in Example 12.1 .
Note that setting up a free-body diagram for a rigid-body equilibrium problem is the most important component in the solution process. Without the correct setup and a correct diagram, you will not be able to write down correct conditions for equilibrium. Also note that a free-body diagram for an extended rigid body that may undergo rotational motion is different from a free-body diagram for a body that experiences only translational motion (as you saw in the chapters on Newton’s laws of motion). In translational dynamics, a body is represented as its CM, where all forces on the body are attached and no torques appear. This does not hold true in rotational dynamics, where an extended rigid body cannot be represented by one point alone. The reason for this is that in analyzing rotation, we must identify torques acting on the body, and torque depends both on the acting force and on its lever arm. Here, the free-body diagram for an extended rigid body helps us identify external torques.
Three masses are attached to a uniform meter stick, as shown in Figure \(\PageIndex{1}\). The mass of the meter stick is 150.0 g and the masses to the left of the fulcrum are m 1 = 50.0 g and m 2 = 75.0 g. Find the mass m3 that balances the system when it is attached at the right end of the stick, and the normal reaction force at the fulcrum when the system is balanced.
Strategy
For the arrangement shown in the figure, we identify the following five forces acting on the meter stick:
- w 1 = m 1 g is the weight of mass m 1 ;
- w 2 = m 2 g is the weight of mass m 2 ;
- w = mg is the weight of the entire meter stick;
- w 3 = m 3 g is the weight of unknown mass m 3 ;
- F S is the normal reaction force at the support point S.
We choose a frame of reference where the direction of the y-axis is the direction of gravity, the direction of the xaxis is along the meter stick, and the axis of rotation (the z-axis) is perpendicular to the x-axis and passes through the support point S. In other words, we choose the pivot at the point where the meter stick touches the support. This is a natural choice for the pivot because this point does not move as the stick rotates. Now we are ready to set up the free-body diagram for the meter stick. We indicate the pivot and attach five vectors representing the five forces along the line representing the meter stick, locating the forces with respect to the pivot Figure \(\PageIndex{2}\). At this stage, we can identify the lever arms of the five forces given the information provided in the problem. For the three hanging masses, the problem is explicit about their locations along the stick, but the information about the location of the weight w is given implicitly. The key word here is “uniform.” We know from our previous studies that the CM of a uniform stick is located at its midpoint, so this is where we attach the weight w, at the 50-cm mark.
Solution
With Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\) for reference, we begin by finding the lever arms of the five forces acting on the stick:
\[\begin{split} r_{1} & = 30.0\; cm + 40.0\; cm = 70.0\; cm \\ r_{2} & = 40.0\; cm \\ r & = 50.0\; cm - 30.0\; cm = 20.0\; cm \\ r_{S} & = 0.0\; cm\; (because\; F_{S}\; is\; attached\; at\; the\; pivot) \\ r_{3} & = 30.0\; cm \ldotp \end{split}\]
Now we can find the five torques with respect to the chosen pivot:
\[\begin{split} \tau_{1} & = +r_{1} w_{1} \sin 90^{o} = +r_{1} m_{1} g \quad (counterclockwise\; rotation,\; positive\; sense) \\ \tau_{2} & = +r_{2} w_{2} \sin 90^{o} = +r_{2} m_{2} g \quad (counterclockwise\; rotation,\; positive\; sense) \\ \tau & = +rw \sin 90^{o} = +rmg \quad \quad \quad (gravitational\; torque) \\ \tau_{S} & = r_{S} F_{S} \sin \theta_{S} = 0 \quad \quad \quad \quad \quad (because\; r_{S} = 0\; cm) \\ \tau_{3} & = -r_{3} w_{3} \sin 90^{o} = -r_{3} m_{3} g \quad (counterclockwise\; rotation,\; negative\; sense) \end{split}\]
The second equilibrium condition (equation for the torques) for the meter stick is
\[\tau_{1} + \tau_{2} + \tau + \tau_{S} + \tau_{3} = 0 \ldotp\]
When substituting torque values into this equation, we can omit the torques giving zero contributions. In this way the second equilibrium condition is
\[+r_{1} m_{1} g + r_{2} m_{2} g + rmg - r_{3} m_{3} g = 0 \ldotp \label{12.17}\]
Selecting the +y-direction to be parallel to \(\vec{F}_{S}\), the first equilibrium condition for the stick is
\[-w_{1} - w_{2} - w + F_{S} - w_{3} = 0 \ldotp\]
Substituting the forces, the first equilibrium condition becomes
\[-m_{1} g - m_{2} g - mg + F_{S} - m_{3} g = 0 \ldotp \label{12.18}\]
We solve these equations simultaneously for the unknown values m 3 and F S . In Equation \ref{12.17}, we cancel the g factor and rearrange the terms to obtain
\[r_{3} m_{3} = r_{1} m_{1} + r_{2} m_{2} + rm \ldotp\]
To obtain m 3 we divide both sides by r 3 , so we have
\[\begin{split} m_{3} & = \frac{r_{1}}{r_{3}} m_{1} + \frac{r_{2}}{r_{3}} m_{2} + \frac{r}{r_{3}} m \\ & = \frac{70}{30} (50.0\; g) + \frac{40}{30} (75.0\; g) + \frac{20}{30} (150.0\; g) = 315.0 \left(\dfrac{2}{3}\right)\; g \simeq 317\; g \ldotp \end{split} \label{12.19}\]
To find the normal reaction force, we rearrange the terms in Equation \ref{12.18}, converting grams to kilograms:
\[\begin{split} F_{S} & = (m_{1} + m_{2} + m + m_{3}) g \\ & = (50.0 + 75.0 + 150.0 + 316.7) \times (10^{-3}\; kg) \times (9.8\; m/s^{2}) = 5.8\; N \ldotp \end{split} \label{12.20}\]
Significance
Notice that Equation \ref{12.17} is independent of the value of g. The torque balance may therefore be used to measure mass, since variations in g-values on Earth’s surface do not affect these measurements. This is not the case for a spring balance because it measures the force.
Repeat Example 12.3 using the left end of the meter stick to calculate the torques; that is, by placing the pivot at the left end of the meter stick.
In the next example, we show how to use the first equilibrium condition (equation for forces) in the vector form given by Equation 12.2.9 and Equation 12.2.10 . We present this solution to illustrate the importance of a suitable choice of reference frame. Although all inertial reference frames are equivalent and numerical solutions obtained in one frame are the same as in any other, an unsuitable choice of reference frame can make the solution quite lengthy and convoluted, whereas a wise choice of reference frame makes the solution straightforward. We show this in the equivalent solution to the same problem. This particular example illustrates an application of static equilibrium to biomechanics.
A weightlifter is holding a 50.0-lb weight (equivalent to 222.4 N) with his forearm, as shown in Figure \(\PageIndex{3}\). His forearm is positioned at \(\beta\) = 60° with respect to his upper arm. The forearm is supported by a contraction of the biceps muscle, which causes a torque around the elbow. Assuming that the tension in the biceps acts along the vertical direction given by gravity, what tension must the muscle exert to hold the forearm at the position shown? What is the force on the elbow joint? Assume that the forearm’s weight is negligible. Give your final answers in SI units.
Strategy
We identify three forces acting on the forearm: the unknown force \(\vec{F}\) at the elbow; the unknown tension \(\vec{T}_{M}\) in the muscle; and the weight \(\vec{w}\) with magnitude w = 50 lb. We adopt the frame of reference with the x-axis along the forearm and the pivot at the elbow. The vertical direction is the direction of the weight, which is the same as the direction of the upper arm. The x-axis makes an angle \(\beta\) = 60° with the vertical. The y-axis is perpendicular to the x-axis. Now we set up the free-body diagram for the forearm. First, we draw the axes, the pivot, and the three vectors representing the three identified forces. Then we locate the angle \(\beta\) and represent each force by its x- and y-components, remembering to cross out the original force vector to avoid double counting. Finally, we label the forces and their lever arms. The free-body diagram for the forearm is shown in Figure \(\PageIndex{4}\). At this point, we are ready to set up equilibrium conditions for the forearm. Each force has x- and y-components; therefore, we have two equations for the first equilibrium condition, one equation for each component of the net force acting on the forearm.
Notice that in our frame of reference, contributions to the second equilibrium condition (for torques) come only from the y-components of the forces because the x-components of the forces are all parallel to their lever arms, so that for any of them we have sin \(\theta\) = 0 in Equation 12.2.12 . For the y-components we have \(\theta\) = ± 90° in Equation 12.2.12 . Also notice that the torque of the force at the elbow is zero because this force is attached at the pivot. So the contribution to the net torque comes only from the torques of T y and of w y .
Solution
We see from the free-body diagram that the x-component of the net force satisfies the equation
\[+F_{x} + T_{x} - w_{x} = 0 \label{12.21}\]
and the y-component of the net force satisfies
\[+F_{y} + T_{y} - w_{y} = 0 \ldotp \label{12.22}\]
Equation \ref{12.21} and Equation \ref{12.22} are two equations of the first equilibrium condition (for forces). Next, we read from the free-body diagram that the net torque along the axis of rotation is
\[+r_{T} T_{y} - r_{w} w_{y} = 0 \ldotp \label{12.23}\]
Equation \ref{12.23} is the second equilibrium condition (for torques) for the forearm. The free-body diagram shows that the lever arms are r T = 1.5 in. and r w = 13.0 in. At this point, we do not need to convert inches into SI units, because as long as these units are consistent in Equation 12.23, they cancel out. Using the free-body diagram again, we find the magnitudes of the component forces:
\[\begin{split} F_{x} & = F \cos \beta = F \cos 60^{o} = \frac{F}{2} \\ T_{x} & = T \cos \beta = T \cos 60^{o} = \frac{T}{2} \\ w_{x} & = w \cos \beta = w \cos 60^{o} = \frac{w}{2} \\ F_{y} & = F \sin \beta = F \sin 60^{o} = \frac{F \sqrt{3}}{2} \\ T_{y} & = T \sin \beta = T \sin 60^{o} = \frac{T \sqrt{3}}{2} \\ w_{y} & = w \sin \beta = w \sin 60^{o} = \frac{w \sqrt{3}}{2} \ldotp \end{split}\]
We substitute these magnitudes into Equation \ref{12.21}, Equation \ref{12.22}, and Equation \ref{12.23} to obtain, respectively,
\[\begin{split} \frac{F}{2} + \frac{T}{2} - \frac{w}{2} & = 0 \\ \frac{F \sqrt{3}}{2} + \frac{T \sqrt{3}}{2} - \frac{w \sqrt{3}}{2} & = 0 \\ \frac{r_{T} T \sqrt{3}}{2} - \frac{r_{w} w \sqrt{3}}{2} & = 0 \ldotp \end{split}\]
When we simplify these equations, we see that we are left with only two independent equations for the two unknown force magnitudes, F and T, because Equation \ref{12.21} for the x-component is equivalent to Equation \ref{12.22} for the y-component. In this way, we obtain the first equilibrium condition for forces
\[F + T - w = 0 \label{12.24}\]
and the second equilibrium condition for torques
\[r_{T} T - r_{w} w = 0 \ldotp \label{12.25}\]
The magnitude of tension in the muscle is obtained by solving Equation \ref{12.25}:
\[T = \frac{r_{w}}{r_{T}} w =frac{13.0}{1.5} (50\; lb) = 433 \frac{1}{3}\; lb \simeq 433.3\; lb \ldotp\]
The force at the elbow is obtained by solving Equation \ref{12.24}:
\[F = w - T = 50.0\; lb - 433.3\; lb = -383.3\; lb \ldotp\]
The negative sign in the equation tells us that the actual force at the elbow is antiparallel to the working direction adopted for drawing the free-body diagram. In the final answer, we convert the forces into SI units of force. The answer is
\[F = 383.3\; lb = 383.3(4.448\; N) = 1705\; N\; downward\]
\[T = 433.3\; lb = 433.3(4.448\; N) = 1927\; N\; upward \ldotp\]
Significance
Two important issues here are worth noting. The first concerns conversion into SI units, which can be done at the very end of the solution as long as we keep consistency in units. The second important issue concerns the hinge joints such as the elbow. In the initial analysis of a problem, hinge joints should always be assumed to exert a force in an arbitrary direction , and then you must solve for all components of a hinge force independently. In this example, the elbow force happens to be vertical because the problem assumes the tension by the biceps to be vertical as well. Such a simplification, however, is not a general rule.
Solution 2
Suppose we adopt a reference frame with the direction of the y-axis along the 50-lb weight and the pivot placed at the elbow. In this frame, all three forces have only y-components, so we have only one equation for the first equilibrium condition (for forces). We draw the free-body diagram for the forearm as shown in Figure \(\PageIndex{5}\), indicating the pivot, the acting forces and their lever arms with respect to the pivot, and the angles \(\theta_{T}\) and \(\theta_{w}\) that the forces \(\vec{T}_{M}\) and \(\vec{w}\) (respectively) make with their lever arms. In the definition of torque given by Equation 12.2.12 , the angle \(\theta_{T}\) is the direction angle of the vector \(\vec{T}_{M}\), counted counterclockwise from the radial direction of the lever arm that always points away from the pivot. By the same convention, the angle \(\theta_{w}\) is measured counterclockwise from the radial direction of the lever arm to the vector \(\vec{w}\). Done this way, the non-zero torques are most easily computed by directly substituting into Equation 12.2.12 as follows:
\[\tau_{T} = r_{T} T \sin \theta_{T} = r_{T} T \sin \beta = r_{T} T \sin 60^{o} = + \frac{r_{T} T \sqrt{3}}{2}\]
\[\tau_{w} = r_{w} w \sin \theta_{w} = r_{w} w \sin (\beta + 180^{o}) = -r_{w} w \sin \beta = - \frac{r_{w} w \sqrt{3}}{2} \ldotp\]
The second equilibrium condition, \(\tau_{T}\) + \(\tau_{w}\) = 0, can be now written as
\[\frac{r_{T} T \sqrt{3}}{2} - \frac{r_{w} w \sqrt{3}}{2} = 0 \ldotp \label{12.26}\]
From the free-body diagram, the first equilibrium condition (for forces) is
\[-F + T - w = 0 \ldotp \label{12.27}\]
Equation \ref{12.26} is identical to Equation \ref{12.25} and gives the result T = 433.3 lb. Equation \ref{12.27} gives
\[F = T - w = 433.3\; lb - 50.0\; lb = 383.3\; lb\]
We see that these answers are identical to our previous answers, but the second choice for the frame of reference leads to an equivalent solution that is simpler and quicker because it does not require that the forces be resolved into their rectangular components.
Repeat Example 12.4 assuming that the forearm is an object of uniform density that weighs 8.896 N.
A uniform ladder is L = 5.0 m long and weighs 400.0 N. The ladder rests against a slippery vertical wall, as shown in Figure \(\PageIndex{6}\). The inclination angle between the ladder and the rough floor is \(\beta\) = 53°. Find the reaction forces from the floor and from the wall on the ladder and the coefficient of static friction \(\mu_{s}\) at the interface of the ladder with the floor that prevents the ladder from slipping.
Strategy
We can identify four forces acting on the ladder. The first force is the normal reaction force N from the floor in the upward vertical direction. The second force is the static friction force f = \(\mu_{s}\)N directed horizontally along the floor toward the wall—this force prevents the ladder from slipping. These two forces act on the ladder at its contact point with the floor. The third force is the weight w of the ladder, attached at its CM located midway between its ends. The fourth force is the normal reaction force F from the wall in the horizontal direction away from the wall, attached at the contact point with the wall. There are no other forces because the wall is slippery, which means there is no friction between the wall and the ladder. Based on this analysis, we adopt the frame of reference with the y-axis in the vertical direction (parallel to the wall) and the x-axis in the horizontal direction (parallel to the floor). In this frame, each force has either a horizontal component or a vertical component but not both, which simplifies the solution. We select the pivot at the contact point with the floor. In the free-body diagram for the ladder, we indicate the pivot, all four forces and their lever arms, and the angles between lever arms and the forces, as shown in Figure \(\PageIndex{7}\). With our choice of the pivot location, there is no torque either from the normal reaction force N or from the static friction f because they both act at the pivot.
Solution
From the free-body diagram, the net force in the x-direction is
\[+f - F = 0 \label{12.28}\]
the net force in the y-direction is
\[+ N - w = 0 \label{12.29}\]
and the net torque along the rotation axis at the pivot point is
\[\tau_{w} + \tau_{F} = 0 \ldotp \label{12.30}\]
where \(\tau_{w}\) is the torque of the weight w and \(\tau_{F}\) is the torque of the reaction F. From the free-body diagram, we identify that the lever arm of the reaction at the wall is r F = L = 5.0 m and the lever arm of the weight is r w = \(\frac{L}{2}\) = 2.5 m. With the help of the free-body diagram, we identify the angles to be used in Equation 12.2.12 for torques: \(\theta_{F}\) = 180° − \(\beta\) for the torque from the reaction force with the wall, and \(\theta_{w}\) = 180° + (90° − \(\beta\)) for the torque due to the weight. Now we are ready to use Equation 12.2.12 to compute torques:
\[\tau_{w} = r_{w} w \sin \theta_{w} = r_{w} w \sin (180^{o} + 90^{o} - \beta) = - \frac{L}{2} w \sin (90^{o} - \beta) = - \frac{L}{2} w \cos \beta\]
\[tau_{F} = r_{F} F \sin \theta_{F} = r_{F} F \sin (180^{o} - \beta) = LF \sin \beta \ldotp\]
We substitute the torques into Equation \ref{12.30} and solve for F :
\[- \frac{L}{2} w \cos \beta + LF \sin \beta = 0 \label{12.31}\]
\[F = \frac{w}{2} \cot \beta = \frac{400.0\; N}{2} \cot 53^{o} = 150.7\; N\]
We obtain the normal reaction force with the floor by solving Equation \ref{12.29}: N = w = 400.0 N. The magnitude of friction is obtained by solving Equation \ref{12.28}: f = F = 150.7 N. The coefficient of static friction is \(\mu_{s}\) = \(\frac{f}{N}\) = \(\frac{150.7}{400.0}\) = 0.377.
The net force on the ladder at the contact point with the floor is the vector sum of the normal reaction from the floor and the static friction forces:
\[\vec{F}_{floor} = \vec{f} + \vec{N} = (150.7\; N)(- \hat{i}) + (400.0\; N)(+ \hat{j}) = (-150.7\; \hat{i} + 400.0\; \hat{j}) N \ldotp\]
Its magnitude is
\[F_{floor} = \sqrt{f^{2} + N^{2}} = \sqrt{150.7^{2} + 400.0^{2}}\; N = 427.4\; N\]
and its direction is
\[\varphi = \tan^{-1} \left(\dfrac{N}{f}\right) = \tan^{-1} \left(\dfrac{400.0}{150.7}\right) = 69.3^{o}\; above\; the\; floor \ldotp\]
We should emphasize here two general observations of practical use. First, notice that when we choose a pivot point, there is no expectation that the system will actually pivot around the chosen point. The ladder in this example is not rotating at all but firmly stands on the floor; nonetheless, its contact point with the floor is a good choice for the pivot. Second, notice when we use Equation 12.2.12 for the computation of individual torques, we do not need to resolve the forces into their normal and parallel components with respect to the direction of the lever arm, and we do not need to consider a sense of the torque. As long as the angle in Equation 12.2.12 is correctly identified—with the help of a free-body diagram—as the angle measured counterclockwise from the direction of the lever arm to the direction of the force vector, Equation 12.2.12 gives both the magnitude and the sense of the torque. This is because torque is the vector product of the lever-arm vector crossed with the force vector, and Equation 12.2.12 expresses the rectangular component of this vector product along the axis of rotation.
Significance
This result is independent of the length of the ladder because L is canceled in the second equilibrium condition, Equation \ref{12.31}. No matter how long or short the ladder is, as long as its weight is 400 N and the angle with the floor is 53°, our results hold. But the ladder will slip if the net torque becomes negative in Equation \ref{12.31}. This happens for some angles when the coefficient of static friction is not great enough to prevent the ladder from slipping.
For the situation described in Example 12.5, determine the values of the coefficient \(\mu_{s}\) of static friction for which the ladder starts slipping, given that β is the angle that the ladder makes with the floor.
A swinging door that weighs w = 400.0 N is supported by hinges A and B so that the door can swing about a vertical axis passing through the hinges Figure \(\PageIndex{8}\). The door has a width of b = 1.00 m, and the door slab has a uniform mass density. The hinges are placed symmetrically at the door’s edge in such a way that the door’s weight is evenly distributed between them. The hinges are separated by distance a = 2.00 m. Find the forces on the hinges when the door rests half-open.
Strategy
The forces that the door exerts on its hinges can be found by simply reversing the directions of the forces that the hinges exert on the door. Hence, our task is to find the forces from the hinges on the door. Three forces act on the door slab: an unknown force \(\vec{A}\) from hinge A, an unknown force \(\vec{B}\) from hinge B, and the known weight \(\vec{w}\) attached at the center of mass of the door slab. The CM is located at the geometrical center of the door because the slab has a uniform mass density. We adopt a rectangular frame of reference with the y-axis along the direction of gravity and the x-axis in the plane of the slab, as shown in panel (a) of Figure \(\PageIndex{9}\), and resolve all forces into their rectangular components. In this way, we have four unknown component forces: two components of force \(\vec{A}\) (A x and A y ), and two components of force \(\vec{B}\) (B x and B y ). In the free-body diagram, we represent the two forces at the hinges by their vector components, whose assumed orientations are arbitrary. Because there are four unknowns (A x , B x , A y , and B y ), we must set up four independent equations. One equation is the equilibrium condition for forces in the x-direction. The second equation is the equilibrium condition for forces in the y-direction. The third equation is the equilibrium condition for torques in rotation about a hinge. Because the weight is evenly distributed between the hinges, we have the fourth equation, A y = B y . To set up the equilibrium conditions, we draw a free-body diagram and choose the pivot point at the upper hinge, as shown in panel (b) of Figure \(\PageIndex{9}\). Finally, we solve the equations for the unknown force components and find the forces.
Solution
From the free-body diagram for the door we have the first equilibrium condition for forces:
in the x-direction, $$-A_{x} + B_{x} = 0 \Rightarrow A_{x} + B_{x}$$in y-direction, $$+ A_{y} + B_{y} - w = 0 \Rightarrow A_{y} = B_{y} = \frac{w}{2} = \frac{400.0\; N}{2} = 200.0\; N \ldotp\]
We select the pivot at point P (upper hinge, per the free-body diagram) and write the second equilibrium condition for torques in rotation about point P:
pivot at P: $$\tau_{w} + \tau_{Bx} + \tau_{By} = 0 \ldotp \label{12.32}\]
We use the free-body diagram to find all the terms in this equation:
\[\begin{split} \tau_{w} & = dw \sin (- \beta) = -dw \sin \beta = -dw \frac{\frac{b}{2}}{d} = -w \frac{b}{2} \\ \tau_{Bx} & = a B_{x} \sin 90^{o} = + a B_{x} \\ \tau_{By} & = a B_{y} \sin 180^{o} = 0 \ldotp \end{split}\]
In evaluating sin \(\beta\), we use the geometry of the triangle shown in part (a) of the figure. Now we substitute these torques into Equation \ref{12.32} and compute B x :
pivot at P: $$-w \frac{b}{2} + a B_{x} = 0 \Rightarrow B_{x} = w \frac{b}{2a} = (400.0\; N) \frac{1}{2\; \cdotp 2} = 100.0\; N \ldotp\]
Therefore the magnitudes of the horizontal component forces are A x = B x = 100.0 N. The forces on the door are
at the upper hinge: $$\vec{F}_{A\; on\; door} = -100.0\; N\; \hat{i} + 200.0\; N\; \hat{j}$$at the lower hinge: $$\vec{F}_{B\; on\; door} = +100.0\; N\; \hat{i} + 200.0\; N\; \hat{j} \ldotp\]
The forces on the hinges are found from Newton’s third law as
on the upper hinge: $$\vec{F}_{door\; on\; A} = 100.0\; N\; \hat{i} - 200.0\; N\; \hat{j}$$on the lower hinge: $$\vec{F}_{door\; on\; B} = - 100.0\; N\; \hat{i} - 200.0\; N\; \hat{j} \ldotp\]
Significance
Note that if the problem were formulated without the assumption of the weight being equally distributed between the two hinges, we wouldn’t be able to solve it because the number of the unknowns would be greater than the number of equations expressing equilibrium conditions.
Solve the problem in Example 12.6 by taking the pivot position at the center of mass.
A 50-kg person stands 1.5 m away from one end of a uniform 6.0-m-long scaffold of mass 70.0 kg. Find the tensions in the two vertical ropes supporting the scaffold.
A 400.0-N sign hangs from the end of a uniform strut. The strut is 4.0 m long and weighs 600.0 N. The strut is supported by a hinge at the wall and by a cable whose other end is tied to the wall at a point 3.0 m above the left end of the strut. Find the tension in the supporting cable and the force of the hinge on the strut.


6.4: Writing Lewis Structures for Covalent Compounds
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Learning Objectives
- Draw Lewis structures for covalent compounds.
The following procedure can be used to construct Lewis electron structures for more complex molecules and ions.
How-to: Constructing Lewis electron structures
1. Determine the total number of valence electrons in the molecule or ion.
- Add together the valence electrons from each atom. (Recall that the number of valence electrons is indicated by the position of the element in the periodic table.)
- If the species is a polyatomic ion, remember to add or subtract the number of electrons necessary to give the total charge on the ion.
For CO 3 2 − , for example, we add two electrons to the total because of the −2 charge.
2. Arrange the atoms to show specific connections.
- When there is a central atom, it is usually the least electronegative element in the compound. Chemists usually list this central atom first in the chemical formula (as in CCl 4 and CO 3 2 − , which both have C as the central atom), which is another clue to the compound’s structure.
- Hydrogen and the halogens are almost always connected to only one other atom, so they are usually terminal rather than central.
3. Place a bonding pair of electrons between each pair of adjacent atoms to give a single bond.
- In H 2 O, for example, there is a bonding pair of electrons between oxygen and each hydrogen.
4. Beginning with the terminal atoms, add enough electrons to each atom to give each atom an octet (two for hydrogen).
- These electrons will usually be lone pairs.
5. If any electrons are left over, place them on the central atom.
- We will explain later that some atoms are able to accommodate more than eight electrons.
6. If the central atom has fewer electrons than an octet, use lone pairs from terminal atoms to form multiple (double or triple) bonds to the central atom to achieve an octet.
- This will not change the number of electrons on the terminal atoms.
7. Final check
- Always make sure all valence electrons are accounted for and that each atom has an octet of electrons, except for hydrogen (with two electrons).
- The central atom is usually the least electronegative element in the molecule or ion; hydrogen and the halogens are usually terminal.
Now let’s apply this procedure to some particular compounds, beginning with one we have already discussed.
Example \(\PageIndex{1}\): Water
Write the Lewis Structure for H 2 O.
Solution
| Steps for Writing Lewis Structures | Example \(\PageIndex{1}\) |
|---|---|
| 1. Determine the total number of valence electrons in the molecule or ion. | Each H atom (group 1) has 1 valence electron, and the O atom (group 16) has 6 valence electrons, for a total of 8 valence electrons. |
| 2. Arrange the atoms to show specific connections. |
H O HBecause H atoms are almost always terminal, the arrangement within the molecule must be HOH . |
|
3. Place a bonding pair of electrons between each pair of adjacent atoms to give a single bond. 4. Beginning with the terminal atoms, add enough electrons to each atom to give each atom an octet (two for hydrogen). |
Placing one bonding pair of electrons between the O atom and each H atom gives H -O- Hwith 4 electrons left over. Each H atom has a full valence shell of 2 electrons. |
| 5. If any electrons are left over, place them on the central atom. |
Adding the remaining 4 electrons to the oxygen (as two lone pairs) gives the following structure:
|
| 6. If the central atom has fewer electrons than an octet, use lone pairs from terminal atoms to form multiple (double or triple) bonds to the central atom to achieve an octet. | Not necessary. |
| 7. Final check. | The Lewis structure gives oxygen an octet and each hydrogen 2 electrons. |
Example \(\PageIndex{2}\)
Write the Lewis structure for the \(CH_2O\) molecule
Solution
| Steps for Writing Lewis Structures | Example \(\PageIndex{2}\) |
|---|---|
| 1. Determine the total number of valence electrons in the molecule or ion. | Each hydrogen atom (group 1) has 1 valence electron, carbon (group 14) has 4 valence electrons, and oxygen (group 16) has 6 valence electrons, for a total of [(2)(1) + 4 + 6] = 12 valence electrons. |
| 2. Arrange the atoms to show specific connections. |
Because carbon is less electronegative than oxygen and hydrogen is normally terminal, C must be the central atom. |
|
3. Place a bonding pair of electrons between each pair of adjacent atoms to give a single bond. |
Placing a bonding pair of electrons between each pair of bonded atoms gives the following:
6 electrons are used, and 6 are left over.
|
| 4. Beginning with the terminal atoms, add enough electrons to each atom to give each atom an octet (two for hydrogen). |
Adding all 6 remaining electrons to oxygen (as three lone pairs) gives the following: Although oxygen now has an octet and each hydrogen has 2 electrons, carbon has only 6 electrons. |
| 5. If any electrons are left over, place them on the central atom. |
Not necessary. There are no electrons left to place on the central atom. |
| 6. If the central atom has fewer electrons than an octet, use lone pairs from terminal atoms to form multiple (double or triple) bonds to the central atom to achieve an octet. |
To give carbon an octet of electrons, we use one of the lone pairs of electrons on oxygen to form a carbon–oxygen double bond: |
| 7. Final check | Both the oxygen and the carbon now have an octet of electrons, so this is an acceptable Lewis electron structure. The O has two bonding pairs and two lone pairs, and C has four bonding pairs. This is the structure of formaldehyde, which is used in embalming fluid. |
Exercise \(\PageIndex{1}\)
Write Lewis electron structures for CO 2 and SCl 2 , a vile-smelling, unstable red liquid that is used in the manufacture of rubber.
- Answer CO 2
-
.
- Answer SCl 2
-
.
The United States Supreme Court has the unenviable task of deciding what the law is. This responsibility can be a major challenge when there is no clear principle involved or where there is a new situation not encountered before. Chemistry faces the same challenge in extending basic concepts to fit a new situation. Drawing of Lewis structures for polyatomic ions uses the same approach, but tweaks the process a little to fit a somewhat different set of circumstances.
Writing Lewis Structures for Polyatomic Ions (CK-12)
Recall that a polyatomic ion is a group of atoms that are covalently bonded together and which carry an overall electrical charge. The ammonium ion, \(\ce{NH_4^+}\), is formed when a hydrogen ion \(\left( \ce{H^+} \right)\) attaches to the lone pair of an ammonia \(\left( \ce{NH_3} \right)\) molecule in a coordinate covalent bond.
When drawing the Lewis structure of a polyatomic ion, the charge of the ion is reflected in the number of total valence electrons in the structure. In the case of the ammonium ion:
\(1 \: \ce{N}\) atom \(= 5\) valence electrons
\(4 \: \ce{H}\) atoms \(= 4 \times 1 = 4\) valence electrons
subtract 1 electron for the \(1+\) charge of the ion
total of 8 valence electrons in the ion
It is customary to put the Lewis structure of a polyatomic ion into a large set of brackets, with the charge of the ion as a superscript outside of the brackets.
Exercise \(\PageIndex{2}\)
Draw the Lewis electron dot structure for the sulfate ion.
- Answer ( CK12 License)
Exceptions to the Octet Rule (BC Campus)
As important and useful as the octet rule is in chemical bonding, there are some well-known violations. This does not mean that the octet rule is useless—quite the contrary. As with many rules, there are exceptions, or violations.
There are three violations to the octet rule. Odd-electron molecules represent the first violation to the octet rule. Although they are few, some stable compounds have an odd number of electrons in their valence shells. With an odd number of electrons, at least one atom in the molecule will have to violate the octet rule. Examples of stable odd-electron molecules are NO , NO 2 , and ClO 2 . The Lewis electron dot diagram for NO is as follows:
Although the O atom has an octet of electrons, the N atom has only seven electrons in its valence shell. Although NO is a stable compound, it is very chemically reactive, as are most other odd-electron compounds.
Electron-deficient molecules represent the second violation to the octet rule. These stable compounds have less than eight electrons around an atom in the molecule. The most common examples are the covalent compounds of beryllium and boron. For example, beryllium can form two covalent bonds, resulting in only four electrons in its valence shell:
Boron commonly makes only three covalent bonds, resulting in only six valence electrons around the B atom. A well-known example is BF 3 :
The third violation to the octet rule is found in those compounds with more than eight electrons assigned to their valence shell. These are called expanded valence shell molecules. Such compounds are formed only by central atoms in the third row of the periodic table or beyond that have empty d orbitals in their valence shells that can participate in covalent bonding. One such compound is PF 5 . The only reasonable Lewis electron dot diagram for this compound has the P atom making five covalent bonds:
Formally, the P atom has 10 electrons in its valence shell.
Example \(\PageIndex{3}\): Octet Violations
Identify each violation to the octet rule by drawing a Lewis electron dot diagram.
- ClO
- SF 6
Solution
a. With one Cl atom and one O atom, this molecule has 6 + 7 = 13 valence electrons, so it is an odd-electron molecule. A Lewis electron dot diagram for this molecule is as follows:
b. In SF 6 , the central S atom makes six covalent bonds to the six surrounding F atoms, so it is an expanded valence shell molecule. Its Lewis electron dot diagram is as follows:
Exercise \(\PageIndex{3}\): Xenon Difluoride
Identify the violation to the octet rule in XeF 2 by drawing a Lewis electron dot diagram.
- Answer
-
The Xe atom has an expanded valence shell with more than eight electrons around it.
Summary
Lewis dot symbols provide a simple rationalization of why elements form compounds with the observed stoichiometries. A plot of the overall energy of a covalent bond as a function of internuclear distance is identical to a plot of an ionic pair because both result from attractive and repulsive forces between charged entities. In Lewis electron structures, we encounter bonding pairs , which are shared by two atoms, and lone pairs , which are not shared between atoms. Lewis structures for polyatomic ions follow the same rules as those for other covalent compounds. There are three violations to the octet rule: odd-electron molecules, electron-deficient molecules, and expanded valence shell molecules.


7.5: Data Validation
In computer science , data validation is the process of ensuring data has undergone data cleansing to ensure they have data quality , that is, that they are both correct and useful. It uses routines, often called "validation rules", "validation constraints", or "check routines", that check for correctness, meaningfulness, and security of data that are input to the system. The rules may be implemented through the automated facilities of a data dictionary , or by the inclusion of explicit application program validation logic of the computer and its application.
This is distinct from formal verification , which attempts to prove or disprove the correctness of algorithms for implementing a specification or property.
Overview
Data validation is intended to provide certain well-defined guarantees for fitness and consistency of data in an application or automated system. Data validation rules can be defined and designed using various methodologies, and be deployed in various contexts. [1] Their implementation can use declarative data integrity rules, or procedure-based business rules . [2]
Note that the guarantees of data validation do not necessarily include accuracy, and it is possible for data entry errors such as misspellings to be accepted as valid. Other clerical and/or computer controls may be applied to reduce inaccuracy within a system.
Different kinds
In evaluating the basics of data validation, generalizations can be made regarding the different kinds of validation according to their scope, complexity, and purpose.
For example:
- Data type validation;
- Range and constraint validation;
- Code and cross-reference validation;
- Structured validation; and
- Consistency validation
Data-type check
Data type validation is customarily carried out on one or more simple data fields.
The simplest kind of data type validation verifies that the individual characters provided through user input are consistent with the expected characters of one or more known primitive data types as defined in a programming language or data storage and retrieval mechanism.
For example, an integer field may require input to use only characters 0 through 9.
Simple range and constraint check
Simple range and constraint validation may examine input for consistency with a minimum/maximum range, or consistency with a test for evaluating a sequence of characters, such as one or more tests against regular expressions. For example, a counter value may be required to be a non-negative integer, and a password may be required to meet a minimum length and contain characters from multiple categories.
Code and cross-reference check
Code and cross-reference validation includes operations to verify that data is consistent with one or more possibly-external rules, requirements, or collections relevant to a particular organization, context or set of underlying assumptions. These additional validity constraints may involve cross-referencing supplied data with a known look-up table or directory information service such as LDAP .
For example, a user-provided country code might be required to identify a current geopolitical region.
Structured check
Structured validation allows for the combination of other kinds of validation, along with more complex processing. Such complex processing may include the testing of conditional constraints for an entire complex data object or set of process operations within a system.
Consistency check
Consistency validation ensures that data is logical. For example, the delivery date of an order can be prohibited from preceding its shipment date.
Example
Multiple kinds of data validation are relevant to 10-digit pre-2007 ISBNs (the 2005 edition of ISO 2108 required ISBNs to have 13 digits from 2007 onwards [3] ).
- Size. A pre-2007 ISBN must consist of 10 digits, with optional hyphens or spaces separating its four parts.
- Format checks. Each of the first 9 digits must be 0 through 9, and the 10th must be either 0 through 9 or an X .
- Check digit . To detect transcription errors in which digits have been altered or transposed, the last digit of a pre-2007 ISBN must match the result of a mathematical formula incorporating the other 9 digits ( ISBN-10 check digits ).
Validation Types
- Allowed character checks
- Checks to ascertain that only expected characters are present in a field. For example a numeric field may only allow the digits 0–9, the decimal point and perhaps a minus sign or commas. A text field such as a personal name might disallow characters used for markup . An e-mail address might require at least one @ sign and various other structural details. Regular expressions can be effective ways to implement such checks.
- Batch totals
- Checks for missing records. Numerical fields may be added together for all records in a batch. The batch total is entered and the computer checks that the total is correct, e.g., add the 'Total Cost' field of a number of transactions together.
- Cardinality check
- Checks that record has a valid number of related records. For example, if a contact record is classified as "customer" then it must have at least one associated order (cardinality > 0). This type of rule can be complicated by additional conditions. For example, if a contact record in a payroll database is classified as "former employee" then it must not have any associated salary payments after the separation date (cardinality = 0).
- Check digits
- Used for numerical data. To support error detection, an extra digit is added to a number which is calculated from the other digits.
- Consistency checks
- Checks fields to ensure data in these fields correspond, e.g., if expiration date is in the past then status is not "active".
- Cross-system consistency checks
- Compares data in different systems to ensure it is consistent. Systems may represent the same data differently, in which case comparison requires transformation (e.g., one system may store customer name in a single Name field as 'Doe, John Q', while another uses First_Name 'John' and Last_Name 'Doe' and Middle_Name 'Quality').
- Data type checks
- Checks input conformance with typed data. For example, an input box accepting numeric data may reject the letter 'O'.
- File existence check
- Checks that a file with a specified name exists. This check is essential for programs that use file handling.
- Format check
- Checks that the data is in a specified format (template), e.g., dates have to be in the format YYYY-MM-DD. Regular expressions may be used for this kind of validation.
- Presence check
- Checks that data is present, e.g., customers may be required to have an email address.
- Range check
- Checks that the data is within a specified range of values, e.g., a probability must be between 0 and 1.
- Referential integrity
- Values in two relational database tables can be linked through foreign key and primary key. If values in the foreign key field are not constrained by internal mechanisms, then they should be validated to ensure that the referencing table always refers to a row in the referenced table.
- Spelling and grammar check
- Looks for spelling and grammatical errors.
- Uniqueness check
- Checks that each value is unique. This can be applied to several fields (i.e. Address, First Name, Last Name).
- Table look up check
- A table look up check compares data to a collection of allowed values.
Source - Wikipedia


8.E: Estimation (Exercises)
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You may want to use the Analysis Lab and various calculators for some of these exercises.
Calculators:
- Inverse t Distribution: Finds t for a confidence interval.
- t Distribution: Computes areas of the t distribution.
- Fisher's r to z': Computes transformations in both directions.
- Inverse Normal Distribution: Use for confidence intervals.
General Questions
Q1
When would the mean grade in a class on a final exam be considered a statistic? When would it be considered a parameter? (relevant section)
Q2
Define bias in terms of expected value. (relevant section)
Q3
Is it possible for a statistic to be unbiased yet very imprecise? How about being very accurate but biased? (relevant section)
Q4
Why is a \(99\%\) confidence interval wider than a \(95\%\) confidence interval? (relevant section & relevant section)
Q5
When you construct a \(95\%\) confidence interval, what are you \(95\%\) confident about? (relevant section)
Q6
What is the difference in the computation of a confidence interval between cases in which you know the population standard deviation and cases in which you have to estimate it? (relevant section & relevant section)
Q7
Assume a researcher found that the correlation between a test he or she developed and job performance was \(0.55\) in a study of \(28\) employees. If correlations under \(0.35\) are considered unacceptable, would you have any reservations about using this test to screen job applicants? (relevant section)
Q8
What is the effect of sample size on the width of a confidence interval? (relevant section & relevant section)
Q9
How does the \(t\) distribution compare with the normal distribution? How does this difference affect the size of confidence intervals constructed using \(z\) relative to those constructed using \(t\)? Does sample size make a difference? (relevant section)
Q10
The effectiveness of a blood-pressure drug is being investigated. How might an experimenter demonstrate that, on average, the reduction in systolic blood pressure is \(20\) or more? ( relevant section & relevant section)
Q11
A population is known to be normally distributed with a standard deviation of \(2.8\).
- Compute the \(95\%\) confidence interval on the mean based on the following sample of nine: \(8, 9, 10, 13, 14, 16, 17, 20, 21\).
- Now compute the \(99\%\) confidence interval using the same data. (relevant section)
Q12
A person claims to be able to predict the outcome of flipping a coin. This person is correct \(16/25\) times. Compute the \(95\%\) confidence interval on the proportion of times this person can predict coin flips correctly. What conclusion can you draw about this test of his ability to predict the future? (relevant section)
Q13
What does it mean that the variance (computed by dividing by \(N\)) is a biased statistic? (relevant section)
Q14
A confidence interval for the population mean computed from an \(N\) of \(16\) ranges from \(12\) to \(28\). A new sample of \(36\) observations is going to be taken. You can't know in advance exactly what the confidence interval will be because it depends on the random sample. Even so, you should have some idea of what it will be. Give your best estimation. (relevant section)
Q15
You take a sample of \(22\) from a population of test scores, and the mean of your sample is \(60\).
- You know the standard deviation of the population is \(10\). What is the \(99\%\) confidence interval on the population mean.
- Now assume that you do not know the population standard deviation, but the standard deviation in your sample is \(10\). What is the \(99\%\) confidence interval on the mean now? (relevant section)
Q16
You read about a survey in a newspaper and find that \(70\%\) of the \(250\) people sampled prefer Candidate \(A\). You are surprised by this survey because you thought that more like \(50\%\) of the population preferred this candidate. Based on this sample, is \(50\%\) a possible population proportion? Compute the \(95\%\) confidence interval to be sure. (relevant section)
Q17
Heights for teenage boys and girls were calculated. The mean height for the sample of \(12\) boys was \(174\) cm and the variance was \(62\). For the sample of \(12\) girls, the mean was \(166\) cm and the variance was \(65\).
- What is the \(95\%\) confidence interval on the difference between population means?
- What is the \(99\%\) confidence interval on the difference between population means?
- Do you think the mean difference in the population could be about \(5\)? Why or why not? (relevant section)
Q18
You were interested in how long the average psychology major at your college studies per night, so you asked \(10\) psychology majors to tell you the amount they study. They told you the following times: \(2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4\).
- Find the \(95\%\) confidence interval on the population mean.
- Find the \(90\%\) confidence interval on the population mean. (relevant section)
Q19
True/false: As the sample size gets larger, the probability that the confidence interval will contain the population mean gets higher. (relevant section & relevant section)
Q20
True/false: You have a sample of \(9\) men and a sample of \(8\) women. The degrees of freedom for the \(t\) value in your confidence interval on the difference between means is \(16\). (relevant section & relevant section)
Q21
True/false: Greek letters are used for statistics as opposed to parameters. (relevant section)
Q22
True/false: In order to construct a confidence interval on the difference between means, you need to assume that the populations have the same variance and are both normally distributed. (relevant section)
Q23
True/false: The red distribution represents the \(t\) distribution and the blue distribution represents the normal distribution. ( relevant section )
Questions from Case Studies
The following questions are from the Angry Moods (AM) case study.
Q24
(AM#6c) Is there a difference in how much males and females use aggressive behavior to improve an angry mood? For the "Anger-Out" scores, compute a \(99\%\) confidence interval on the difference between gender means. (relevant section)
Q25
(AM#10) Calculate the \(95\%\) confidence interval for the difference between the mean Anger-In score for the athletes and non-athletes. What can you conclude? (relevant section)
Q26
Find the \(95\%\) confidence interval on the population correlation between the Anger-Out and Control-Out scores. (relevant section)
The following questions are from the Flatulence (F) case study.
Q27
(F#8) Compare men and women on the variable "perday." Compute the \(95\%\) confidence interval on the difference between means. (relevant section)
Q28
(F#10) What is the \(95\%\) confidence interval of the mean time people wait before farting in front of a romantic partner. (relevant section)
The following questions use data from the Animal Research (AR) case study.
Q29
(AR#3) What percentage of the women studied in this sample strongly agreed (gave a rating of \(7\)) that using animals for research is wrong?
Q30
Use the proportion you computed in #29. Compute the \(95\%\) confidence interval on the population proportion of women who strongly agree that animal research is wrong. (relevant section)
Q31
Compute a \(95\%\) confidence interval on the difference between the gender means with respect to their beliefs that animal research is wrong. (relevant section)
The following question is from the ADHD Treatment (AT) case study.
Q32
(AT#8) What is the correlation between the participants' correct number of responses after taking the placebo and their correct number of responses after taking \(0.60\) mg/kg of MPH? Compute the \(95\%\) confidence interval on the population correlation. (relevant section)
The following question is from the Weapons and Aggression (WA) case study.
Q33
(WA#4) Recall that the hypothesis is that a person can name an aggressive word more quickly if it is preceded by a weapon word prime than if it is preceded by a neutral word prime. The first step in testing this hypothesis is to compute the difference between
- the naming time of aggressive words when preceded by a neutral word prime and
- the naming time of aggressive words when preceded by a weapon word prime separately for each of the \(32\) participants. That is, compute an \(- aw\) for each participant.
- Would the hypothesis of this study be supported if the difference were positive or if it were negative?
- What is the mean of this difference score? (relevant section)
- What is the standard deviation of this difference score? (relevant section)
- What is the 95% confidence interval of the mean difference score? (relevant section)
- What does the confidence interval computed in (d) say about the hypothesis.
The following question is from the Diet and Health (WA) case study.
Q34
Compute a \(95\%\) confidence interval on the proportion of people who are healthy on the AHA diet.
| Cancers | Deaths | Nonfatal illness | Healthy | Total | |
| AHA |
The following questions are from (reproduced with permission) Visit the site
Q35
Suppose that you take a random sample of \(10,000\) Americans and find that \(1,111\) are left-handed. You perform a test of significance to assess whether the sample data provide evidence that more than \(10\%\) of all Americans are left-handed, and you calculate a test statistic of \(3.70\) and a \(p\)-value of \(0.0001\). Furthermore, you calculate a \(99\%\) confidence interval for the proportion of left-handers in America to be \((0.103,0.119)\). Consider the following statements: The sample provides strong evidence that more than \(10\%\) of all Americans are left-handed. The sample provides evidence that the proportion of left-handers in America is much larger than \(10\%\). Which of these two statements is the more appropriate conclusion to draw? Explain your answer based on the results of the significance test and confidence interval.
Q36
A student wanted to study the ages of couples applying for marriage licenses in his county. He studied a sample of \(94\) marriage licenses and found that in \(67\) cases the husband was older than the wife. Do the sample data provide strong evidence that the husband is usually older than the wife among couples applying for marriage licenses in that county? Explain briefly and justify your answer.
Q37
Imagine that there are \(100\) different researchers each studying the sleeping habits of college freshmen. Each researcher takes a random sample of size \(50\) from the same population of freshmen. Each researcher is trying to estimate the mean hours of sleep that freshmen get at night, and each one constructs a \(95\%\) confidence interval for the mean. Approximately how many of these \(100\) confidence intervals will NOT capture the true mean?
- None
- \(1\) or \(2\)
- \(3\) to \(7\)
- about half
- \(95\) to \(100\)
- other
Selected Answers
S11
- \((12.39, 16.05)\)
S12
\((0.43, 0.85)\)
S15
- \((53.96, 66.04)\)
S17
- \((1.25, 14.75)\)
S18
- \((1.45, 3.05)\)
S26
\((-0.713, -0.414)\)
S27
\((-0.98, 3.09)\)
S29
\(41\%\)
S33
- \(7.16\)


2.6E: Exercises
Exercises
Verbal
Algebraic
For the following exercises, use the Remainder Theorem to find the remainder.
For the following exercises, use the given factor and the Factor Theorem to find all real zeros for the given polynomial function.
For the following exercises, use the Rational Zero Theorem to find all real zeros.
For the following exercises, find all complex solutions (real and non-real).
Graphical
For the following exercises, use Descartes’ Rule to determine the possible number of positive and negative solutions. Then graph to confirm which of those possibilities is the actual combination.
Numeric
For the following exercises, list all possible rational zeros for the functions.
Technology
For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.
Extensions
For the following exercises, construct a polynomial function of least degree possible using the given information.
Real-World Applications
For the following exercises, find the dimensions of the box described.
For the following exercises, find the dimensions of the right circular cylinder described.


47.1: More than Two Choices
Lesson
Let's explore different ways to determine a winner.
Exercise \(\PageIndex{1}\): Field Day
Students in a sixth-grade class were asked, “What activity would you most like to do for field day?” The results are shown in the table.
| activity | number of votes |
|---|---|
| softball game | \(16\) |
| scavenger hunt | \(10\) |
| dancing talent show | \(8\) |
| marshmallow throw | \(4\) |
| no preference | (2\) |
- What percentage of the class voted for softball?
- What percentage did not vote for softball as their first choice?
Exercise \(\PageIndex{2}\): School Lunches (Part 1)
Suppose students at our school are voting for the lunch menu over the course of one week. The following is a list of options provided by the caterer.
Menu 1: Meat Lovers
- Meat loaf
- Hot dogs
- Pork cutlets
- Beef stew
- Liver and onions
Menu 2: Vegetarian
- Vegetable soup and peanut butter sandwich
- Hummus, pita, and veggie sticks
- Veggie burgers and fries
- Chef’s salad
- Cheese pizza every day
- Double desserts every day
Menu 3: Something for Everyone
- Chicken nuggets
- Burgers and fries
- Pizza
- Tacos
- Leftover day (all the week’s leftovers made into a casserole)
- Bonus side dish: pea jello (green gelatin with canned peas)
Menu 4: Concession Stand
- Choice of hamburger or hot dog, with fries, every day
To vote, draw one of the following symbols next to each menu option to show your first, second, third, and last choices. If you use the slips of paper from your teacher, use only the column that says “symbol.”
- Meat Lovers __________
- Vegetarian __________
- Something for Everyone __________
- Concession Stand __________
Here are two voting systems that can be used to determine the winner.
- Voting System #1. Plurality : The option with the most first-choice votes (stars) wins.
- Voting System #2. Runoff : If no choice received a majority of the votes, leave out the choice that received the fewest first-choice votes (stars). Then have another vote.
If your first vote is still a choice, vote for that. If not, vote for your second choice that you wrote down.
If there is still no majority, leave out the choice that got the fewest votes, and then vote again. Vote for your first choice if it’s still in, and if not, vote for your second choice. If your second choice is also out, vote for your third choice.
- How many people in our class are voting? How many votes does it take to win a majority?
- How many votes did the top option receive? Was this a majority of the votes?
-
People tend to be more satisfied with election results if their top choices win. For how many, and what percentage, of people was the winning option:
- their first choice?
- their second choice?
- their third choice?
- their last choice?
- After the second round of voting, did any choice get a majority? If so, is it the same choice that got a plurality in Voting System #1?
- Which choice won?
-
How satisfied were the voters by the election results? For how many, and what percentage, of people was the winning option:
- their first choice?
- their second choice?
- their third choice?
- their last choice?
- Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?
Exercise \(\PageIndex{3}\): School Lunch (Part 2)
Let’s analyze a different election.
In another class, there are four clubs. Everyone in each club agrees to vote for the lunch menu exactly the same way, as shown in this table.
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Figure out which option won the election by answering these questions.
- On the first vote, when everyone voted for their first choice, how many votes did each option get? Did any choice get a majority?
- Which option is removed from the next vote?
- On the second vote, how many votes did each of the remaining three menu options get? Did any option get a majority?
- Which menu option is removed from the next vote?
- On the third vote, how many votes did each of the remaining two options get? Which option won?
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Estimate how satisfied all the voters were.
- For how many people was the winner their first choice?
- For how many people was the winner their second choice?
- For how many people was the winner their third choice?
- For how many people was the winner their last choice?
- Compare the satisfaction results for the plurality voting rule and the runoff rule. Did one produce satisfactory results for more people than the other?
Exercise \(\PageIndex{4}\): Just Vote Once
Your class just voted using the instant runoff system. Use the class data for following questions.
- For our class, which choice received the most points?
- Does this result agree with that from the runoff election in an earlier activity?
- For the other class, which choice received the most points?
- Does this result agree with that from the runoff election in an earlier activity?
- The runoff method uses information about people’s first, second, third, and last choices when it is not clear that there is a winner from everyone’s first choices. How does the instant runoff method include the same information?
- After comparing the results for the three voting rules (plurality, runoff, instant runoff) and the satisfaction surveys, which method do you think is fairest? Explain.
Are you ready for more?
Numbering your choices 0 through 3 might not really describe your opinions.  You might want to give A and C both a 3, and B and D both a 0.
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Design a numbering system where the size of the number accurately shows how much you like a choice. Some ideas:
- The same 0 to 3 scale, but you can choose more than one of each number, or even decimals between 0 and 3.
- A scale of 1 to 10, with 10 for the best and 1 for the worst.
- Try out your system with the people in your group, using the same school lunch options for the election.
- Do you think your system gives a more fair way to make choices? Explain your reasoning.
Exercise \(\PageIndex{5}\): Weekend Choices
Clare, Han, Mai, Tyler, and Noah are deciding what to do on the weekend. Their options are cooking, hiking, and bowling. Here are the points for their instant runoff vote. Each first choice gets 2 points, the second choice gets 1 point, and the last choice gets 0 points.
| cooking | hiking | bowling | |
|---|---|---|---|
| Clare | \(2\) | \(1\) | \(0\) |
| Han | \(2\) | \(1\) | \(0\) |
| Mai | \(2\) | \(1\) | \(0\) |
| Tyler | \(0\) | \(2\) | \(1\) |
| Noah | \(0\) | \(2\) | \(1\) |
- Which activity won using the instant runoff method? Show your calculations and use expressions or equations.
- Which activity would have won if there was just a vote for their top choice, with a majority or plurality winning?
- Which activity would have won if there was a runoff election?
- Explain why this happened.


1.16: Critical Thinking
- Explore the concept of critical thinking
As a college student, you are tasked with engaging and expanding your thinking skills. One of the most important of these skills is critical thinking. Critical thinking relates to nearly all tasks, situations, topics, careers, environments, challenges, and opportunities—it’s not restricted to a particular subject area.
Critical Thinking
Critical thinking is clear, reasonable, reflective thinking focused on deciding what to believe or do. It means asking probing questions like, “How do we know?” or “Is this true in every case or just in this instance?” It involves being skeptical and challenging assumptions, rather than simply memorizing facts or blindly accepting what you hear or read.
Imagine, for example, that you’re reading a history textbook. You wonder who wrote it and why, because you detect certain assumptions in the writing. You find that the author has a limited scope of research focused only on a particular group within a population. In this case, your critical thinking reveals that there are “other sides to the story.”
Who are critical thinkers, and what characteristics do they have in common? Critical thinkers are usually curious and reflective people. They like to explore and probe new areas and seek knowledge, clarification, and new solutions. They ask pertinent questions, evaluate statements and arguments, and they distinguish between facts and opinion. They do not rely on quick, first-order level thinking, but instead use second-level or higher-order thinking skills that require them to think more deeply before jumping to conclusions. They can evaluate their own opinion and judge whether ideas are their own. They are also willing to examine their own beliefs, possessing a manner of humility that allows them to admit a lack of knowledge or understanding when needed. They are fair-minded and open to changing their mind. Perhaps most of all, they actively enjoy learning, and seeking new knowledge is a lifelong pursuit.
Critical thinking skills are perhaps the most fundamental skills involved in making judgments and solving problems. You use them every day, and you can continue improving them.
The ability to think critically about a matter—to analyze a question, situation, or problem down to its most basic parts—is what helps us evaluate the accuracy and truthfulness of statements, claims, and information we read and hear. It is the sharp knife that, when honed, separates fact from fiction, honesty from lies, and the accurate from the misleading. We all use this skill to one degree or another almost every day.
For example, we use critical thinking every day as we consider the latest consumer products and why one particular product is the best among its peers. Is it a quality product because a celebrity endorses it? Because a lot of other people may have used it? Because it is made by one company versus another? Or perhaps because it is made in one country or another? These are questions representative of critical thinking.
The academic setting demands more of us in terms of critical thinking than everyday life. It demands that we evaluate information and analyze a myriad of issues. It is an environment where our critical thinking skills can be the difference between success and failure. In this environment, we must consider information in an analytical, critical manner. We must ask questions—What is the source of this information? Is this source an expert one and what makes it so? Are there multiple perspectives to consider on an issue? Do multiple sources agree or disagree on an issue? Does quality research substantiate information or opinion? Do I have any personal biases that may affect my consideration of this information?
It is only through purposeful, frequent, intentional questioning such as this that we can sharpen our critical thinking skills and improve as students, learners, and researchers.
—Dr. Andrew Robert Baker, Foundations of Academic Success: Words of Wisdom
No matter where you are on the road to being a critical thinker, you can always more fully develop your skills. Doing so will help you develop more balanced arguments, express yourself clearly, read critically, and absorb important information efficiently. Critical thinking skills will help you in any profession or any circumstance of life, from science to art to business to teaching.
| Critical Thinking IS | Critical Thinking is NOT |
|---|---|
| Skepticism | Memorizing |
| Examining assumptions | Going along with the group |
| Challenging reasoning | Blind acceptance of authority |
| Uncovering biases | Believing everything you read |
You can view the transcript for “Why Critical Thinking (Study Skills)” here (opens in new window) .
Critical thinking is fundamentally a process of questioning. You may question the information you read in a textbook, or you may question what a politician or a professor or a classmate says. You can also question a commonly-held belief or a new idea. With critical thinking, anything and everything is subject to question and examination.
An Example of Critical Thinking
Let’s use a simple example of applying logic to a critical-thinking situation. In this hypothetical scenario, Professor Brown has a PhD in political science, and he works as a professor at a local college. His wife works at the college, too. They have three young children in the local school system, and their family is well known in the community.
Professor Brown is now running for political office. Are his credentials and experience sufficient for entering public office? Will he be effective in the political office? Some voters might believe that his personal life and current job, on the surface, suggest he will do well in the position, and they will vote for him.
In truth, Brown’s characteristics as described above don’t guarantee that he will do a good job. What else might you want to know? How about whether Brown has already held a political office? If so, was he effective? How about Brown’s reputation for personal integrity? Just because he works as a professor does not mean he has experience or skill with the ethical issues of public office. Does he have the leadership skills required to be an effective public leader? Just because he teaches college students does not mean Brown has effective leadership skills. In this case, we want to ask, “How much information is adequate in order to make a decision based on logic instead of assumptions?”
The following questions, presented in Figure 1, below, are ones you may apply to formulate a logical, reasoned perspective in the above scenario or any other situation:
- What’s happening? Gather the basic information and begin to think of questions.
- Why is it important? Ask yourself why it’s significant and whether or not you agree.
- What don’t I see? Is there anything important missing?
- How do I know? Ask yourself where the information came from and how it was constructed.
- Who is saying it? What’s the position of the speaker and what is influencing them?
- What else? What if? What other ideas exist and are there other possibilities?
Problem-Solving With Critical Thinking
Critical thinking is not just an academic exercise. For most people, a typical day is filled with critical thinking. We use critical thinking to solve problems all the time. For example, consider the following situation. Gisella is doing well in college, and most of her college and living expenses are covered, but there are some gaps between what she wants and what she feels she can afford. She uses critical thinking to analyze her income, savings, and budget to better calculate what she will need to stay in college and maintain her desired level of spending.
Remember, when you apply the skills of a good critical thinker to your academic work or to the problems in your life, your challenge will be less of a hurdle. If you are curious, reflective, knowledge-seeking, open to change, probing, organized, and ethical, you’ll be in a good position to find intelligent solutions.
critical thinking : clear, reasonable, reflective thinking focused on deciding what to believe or do
Contributors and Attributions
- Foundations of Academic Success. Authored by : Thomas C. Priester, editor. Provided by : Open SUNY Textbooks. Located at : textbooks.opensuny.org/foundations-of-academic-success/. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Critical Thinking. Provided by : Critical and Creative Thinking Program. Located at : cct.wikispaces.umb.edu/Critical+Thinking. License : CC BY: Attribution
- Critical Thinking Skills. Authored by : Linda Bruce. Provided by : Lumen Learning. Project : courses.lumenlearning.com/lu...inking-skills/. License : CC BY: Attribution
- Why Critical Thinking. Provided by : Careers and Skills. Located at : https://www.youtube.com/watch?v=2ntsfbMXGwU&feature=youtu.be . License : All Rights Reserved . License Terms : Standard YouTube License


21.2C: Larynx
The larynx is an organ in the neck involved in breathing, sound production, and protecting the trachea against food aspiration.
- Describe the anatomic structure of the larynx
Key Points
- In adult humans, the larynx is found in the anterior neck at the level of the C3–C6 vertebrae and consists of nine cartilages: three single (epiglottic, thyroid, and cricoid) and three paired (arytenoid, corniculate, and cuneiform).
- Its interior can be divided in supraglottis, glottis, and subglottis.
- The larynx houses the vocal folds—commonly but improperly termed the vocal cords. They are situated just below where the tract of the pharynx splits into the trachea and the esophagus; they are essential for phonation.
- The vocal folds are closed together by adducting the arytenoid cartilages, so that they vibrate (see phonation).
- The vocal folds are controlled by the action of the vagus nerve.
- The larynx closes and rises during swallowing to move the epiglottis over the trachea.
- The larynx closes during a coughing reflex to protect the lungs from inhaling something that could damage it, and to remove foreign material from the trachea and lungs.
Key Terms
- vocal cords : Two folds of tissue located in the larynx that vibrate when air passes over them, producing the sound waves associated with talking and singing.
- cough reflex : This occurs when the larynx closes as it forces air out of the lungs to protect the lungs from aspirated materials.
- larynges : Part of the respiratory tract between the pharynx and the trachea, having walls of cartilage and muscle and containing the vocal cords enveloped in folds of mucous membrane.
The larynx (plural: larynges), commonly called the voice box, is an organ in the neck of humans and most animals that is involved in breathing, sound production, coughing, and protecting the trachea against food aspiration during eating.
External view of the larynx : This figure is a detailed view of the external aspect of the larynx.
Anatomy of the Larynx
In adult humans, the larynx is found in the anterior neck at the level of the C3–C6 vertebrae in the backbone. It connects the inferior part of the pharynx (laryngopharynx) with the trachea. The laryngeal skeleton consists of three single cartilages (thyroid, epiglottic, and cricoid).
- The thyroid cartilage is particularly notable for forming the Adam’s apple, the visible bulge made by the larynx when looking at the throat, and protects the larynx from injury.
- The epiglottic cartilage is the body of the epiglottis itself that connects to the larynx from above.
- The cricoid cartilage connects the larynx to the trachea from below.
There are also three sets of cartilages that are paired on either side of the larynx (arytenoid, corniculate, and cuneiform) that allow the position of the larynx to move during voice production.
The larynx connects to the hyoid bone (the bone that forms the floor of the mouth) from above. The larynx extends vertically from the tip of the epiglottis to the border of the cricoid cartilage that marks the formal beginning of the trachea.
The interior of the larynx consists of three regions, the supraglottis, glottis, and subglottis. The glottis is the midsection that contains the vocal folds (folds of muscular epithelium ), while the supraglottis and subglottis are the areas of the larynx that are above and below the glottis respectively. In newborn infants, the larynx is initially at the level of the C2–C3 vertebrae, but descends as the child grows.
The glottis consists of two pairs of mucosal folds. These folds are false vocal folds (vestibular folds) and true vocal folds (folds). The false vocal folds are covered by respiratory epithelium, while the true vocal folds are covered by stratified squamous epithelium.
The false vocal folds are not responsible for sound production, but rather for resonance. These false vocal folds do not contain muscle, while the true vocal folds do have skeletal muscle. The two sets of folds are separated by the vocal ligament, with the false vocal folds above, and the true vocal cords below the ligament. The true vocal folds are often referred to as the vocal cords, however the folds technically aren’t cords.
Physiology of the Larynx
The most notable and unique function of the larynx is phonation (voice production). The vocal folds of the larynx have two positions, open and closed. During breathing the folds remain open, but they close during swallowing or phonation.
When air from the lungs passes through closed folds during exhalation, the folds vibrate and create sound. The pitch produced depends on the length and tightness of the vocal folds.
The vagus nerves innervate the larynx and signal the muscles and paired cartilage (the arytenoid) of the larynx to work together to open and close the vocal folds as well as change their length and tension to alter pitch. Longer vocal folds have a lower pitch, which is part of the reason why men have deeper voices compared to women, because their larger larynxes have longer vocal folds.
Besides phonation, there are a few other important functions of the larynx. The folds of the larynx close and move upwards during swallowing, which causes the epiglottis to close off the trachea. This helps prevent aspiration of food into the lungs or choking from a blockage of food in the trachea.
The larynx closes off during coughing to help prevent harmful gasses from entering the lungs. During a cough reflex, air is forced out of the lungs, which can remove accumulated mucus, fluid, or blood from the lungs during injury, infection, or cancer of the lungs, as well as food or objects in the trachea during choking.
Finally, the larynx can be signaled to open its folds wider than usual to increase the flow of air into and out of the lungs during heavy breathing when the body requires more oxygen.


32: How I Celebrate Life on the Day of the Dead (González)
Linda González
#argument, #sharedvalues, #ethos, #pathos, #descriptive, #kairos, #logos, #currentevents, #advice, #artsandculture #global
Every year as November 1 approaches, I do the math to remember how long ago my father passed away on Día de los Muertos. This year, I dutifully pulled up my calculator and subtracted 1996 from 2017. Twenty-one years. And then the obvious hits me. I can always know how long it has been since he passed on to his next life by subtracting one year from my twins’ age. They are 22 and were just a year old when their abuelo died. I remember carrying Gina down the aisle behind the casket, her and Teo’s new life blooming while that same year Tot’s had faded.
I set up my altar this week, pulling out the pictures of my dearly departed and adding new ones from this year. The first step is always laying out the cross-stitched mantle with years of stains and a dark mark from when a candle burned too hot. I tape papel picado above the altar, remembering this ritual is not a dirge; it is an opening of the veil to celebrate the lives that touched me and my comunidades. It is a time to think about why I miss them and ponder how to keep them alive in the present moment.
I imagine my dad’s disappointed spirit hovering over the Dodgers as they lost in the World Series. I invoke my mom’s stovetop magic as I figure out what to do with a bag of zucchini that must be cooked tonight. I remember the mothers who grieve their sons’ vibrant spirits every day, and I take a moment to send Snapchats to my beloved cuates.
Día de los Muertos is so ingrained in my being that I am startled to see people in costume; my mind wonders for a second, “What’s that all about?” This is amazing because I was so involved in Halloween while my children were growing up—making costumes, figuring out the healthiest candy to hand out, trading my children’s candy for money so they were not overloaded with sugar (and I could store their loot for the next Halloween).
In years past, I have hosted gatherings to decorate sugar skulls, loving this tradition of blending death with creativity. I treasured giving my children and their friends the chance to be playful and imaginative with something that so many people fear. As a writer, I live in that crevice of light and shadow, writing drafts only to end their existence for another version and then another and then yet another.
I love the transparency of life and death, the calaveras that dance and meditate and watch TV. Each skeleton could be anyone of us, and one day we will know what our antepasados experienced after their last out-breath. One day we will see there is no separation between any of us, alive and dead.
From the author’s personal altar. Photo from Linda González.
The first and only altar in my parents’ home was the one we created on a cake after my dad’s funeral, laying out the detallitos of his life that he allowed to be visible. The secrets were still within him, wisps of energy that over the years encircled us with cariño or strangled our voices or tripped us as we ran.
As I set up my altar year after year, I breathe in the musty smell of the newspapers I have carried from home to home. These crinkled papelitos wrap and unwrap memories and give space for those I loved and lost to whisper consejos in the stillness. I unbind my heart wounds and apply the salve gained from another year of living—that little bit more of perspective and wisdom nestled in my corazón that wraps around me like a soft, colorful rebozo.
____________________
Linda González is the author of the memoir The Cost of Our Lives. She has published essays in literary journals and books, is a storyteller, and received her MFA from Goddard College. This essay is an excerpt from Endangered Species, Enduring Values: An Anthology of San Francisco Area Writers and Artists of Color , edited by Shizue Seigel, Pease Press, 2018. www.peasepress.com . It was published in Yes! Magazine, 31 October 2018.
How I Celebrate Life on the Day of the Dead by Linda González is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License .


7.3.7B: The Lysogenic Life Cycle of Bacteriophages
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- Ying Liu
- City College of San Francisco
Describe the lysogenic life cycle of temperate phages (including spontaneous induction).
Define the following:
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temperate phage
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lysogen
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prophage
- temperate phage
- lysogen
- prophage
Bacteriophages capable of a lysogenic life cycle are termed temperate bacteriophages. When a temperate bacteriophage infects a bacterium, it can either replicate by means of the lytic life cycle and cause lysis of the host bacterium, or, it can incorporate its DNA into the bacterium's DNA and become a noninfectious prophage (see Figure \(\PageIndex{1}\)) . In the latter case, the cycle begins by the bacteriophage adsorbing to the host bacterium or lysogen and injecting its genome as in the lytic life cycle (see Figure \(\PageIndex{2}\) and Figure \(\PageIndex{3}\) ). However, the bacteriophage does not shut down the host cell. Instead, the bacteriophage DNA inserts or integrates into the host bacterium's DNA (see Figure \(\PageIndex{4}\)) . At this stage the virus is called a prophage. Expression of the bacteriophage genes controlling bacteriophage replication is blocked by a repressor protein, and the phage DNA replicates as a part of the bacterium's DNA so that every daughter bacterium now contains the prophage (see Figure \(\PageIndex{5}\)) .
The number of viruses infecting the bacterium as well as the physiological state of the bacterium appear to determine whether the temperate bacteriophage enters the lytic cycle or becomes a prophage.
In about one out of every million to one out of every billion bacteria containing a prophage, spontaneous induction occurs. The bacteriophage genes are activated and new bacteriophages are produced by the lytic life cycle (see Figure \(\PageIndex{5}\)A , Figure \(\PageIndex{6}\) , Figure \(\PageIndex{7}\) , Figure \(\PageIndex{8}\) , and Figure \(\PageIndex{9}\) ).


2.8: Making an Offer of Employment
After selecting the best candidate, you need to make a job offer. Remember, you are not obliged to hire any of the applicants if you are not satisfied with them. A verbal job offer must be followed by a formal written offer of employment; this ensures that you and the candidate are fully aware of the terms and conditions of employment.
When making a verbal offer, ensure you:
- Congratulate the person on being the successful candidate.
- Enthusiastically welcome the candidate to your team.
- Re-state the position for which the candidate is being hired.
- Inform the candidate of the starting wage you agreed upon, as well as hours/days of work, benefits, vacation time, and any other relevant information
- Confirm the start date and time.
- Inform the candidate of any training programs he or she will be attending on the first day, including length and expectations.
- Inform the candidate of dress code standards.
- Inform the candidate that this information will be put into a formal letter for signing.
Once you have made the verbal offer, provide the candidate with a reasonable time frame to consider the decision and the opportunity to clarify any of the terms of the offer. Be prepared to reconsider your terms and conditions if the candidate asks to negotiate something. Once these details are agreed upon, make sure they are clear in the written offer you provide. The written letter should ask for the candidate’s signature and contain a company representative’s signature. After both of you have signed the letter, give the employee the original and keep a copy in the employee’s file.
Common components of an letter of offer include:
- Position title (attach a copy of the job description)
- Basic duties and responsibilities
- Position status (part-time, full-time, seasonal, temporary full-time)
- Start date
- Performance review expectations
- Salary and overtime
- Work schedule
- Probationary period
- Benefits and insurance entitlement
- Vacation entitlement and statutory holidays
- Union contract (if applicable)
- Training programs
- Bonus plan information, if applicable, and eligibility requirements
- Notice period requirements for termination of employment or resignation
- Reference to an employee handbook and/or any other included attachments/schedules
Don’t forget to include the date the letter is written and the date it is valid until, who it needs to be returned to, in what format, and how. See Appendix 5 for a sample letter of offer.
Informing unsuccessful candidates
Once you have selected and informed the successful applicant, and both of you have signed the letter of offer, you must inform the unsuccessful applicants as quickly as possible so they hear the news from you and not on the street.
The successful applicant is probably ecstatic and might want to tell the world how great your company is without any prompting from you. The unsuccessful people will naturally be disappointed and, unless you handle the situation correctly, they can become very poor ambassadors. Because there are usually many more unsuccessful applicants than successful ones, the potential for spreading bad news about your company is significant.
When informing the unsuccessful applicants, take a forthright and understanding approach. The conversation should be short and to the point. Start by telling them that they were not the successful in their application. You might express the sentiment that they were really good, but the selected person was better qualified. This is true, of course, otherwise they would not have been on the shortlist. Wish the unsuccessful applicants well. For those candidates who stood out, consider asking if you can keep their resumé on file should a similar position become available it the future. Most importantly in every instance, treat the unsuccessful applicants with dignity, respect, and empathy.


12.6E: Finding New Antimicrobial Drugs
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Learning Objectives
- Explain the reasons for low production of new antibiotics and discuss the proposed mechanisms to evade antimicrobial resistance
Antimicrobial resistance: the problem
Antibiotics, more than any other medicines, have improved the life expectancy of mankind, however, multi-drug resistance has become common in pathogenic bacteria and multiple drugs are losing efficacy. Recent reports on the occurrence of panresistant gram-negative strains, i.e. strains resistant to every registered antibacterial drug, indicate that we are on the verge to lose the battle, taking us back to the pre-antibiotic era. There is world-wide consensus that the medical need for novel antiinfective drugs is enormous and that we are running out of time. Many achievements of modern medicine, not only treatment of infectious diseases, depend on the availability of efficacious antibiotics, still, the antibacterial development pipeline is slow and the number of new drugs reaching the market is alarmingly low. There are many reasons for this at all levels of the discovery and development process. Investments into antibiotic research and technologies is minimal; socioeconomic considerations together with regulatory hurdles have prompted pharmaceutical companies to exit the field and innovative biotech companies were confronted with problems beyond their control. Answers are needed as to where and how we can find new lead compounds with unprecedented activities?
Finding new antimicrobial drugs: the solution
Research on new antimicrobial compounds is geared towards innovative targets to circumvent resistance. Some of the proposed areas to investigate include: collecting and examining the list of antimicrobial resistance genes (e.g. exploring the resistome), targeting teichoic acid biosynthesis as a new method to compromise the bacterial wall integrity, producing ribosomal inhibitors to target protein synthesis, targeting outer-membrane transporters with protein epitope mimetics (e.g. mimetics of the cationic antimicrobial peptides that form part of the immune response to microbes), and developing antibody-based strategies and vaccines. The initiative to develop new antimicrobial agents is urgently needed but is a long process from invention, to development, to actual clinical application. It is also necessary to initiate a worldwide awareness on antibiotic misuse and overuse as a mean to address the root of the problem for antimicrobial resistance.
Key Points
- Finding new antimicrobial drugs requires researchers, pharmaceutical, and biotech companies to invest in new technology and discover new sources for antibiotic development.
- Proposed mechanisms to circumvent antimicrobial resistance range from exploring the list of resistance genes to antibody-based therapy and vaccines.
- Finding new candidates to target is essential but it needs to be accompanied by awareness on antibiotic misuse with the prospect to eliminate the root of the problem.
Key Terms
- mimetic : A substance with similar pharmacological effects to another substance.


15: Circulatory and Lymphatic System Infections (Part A)
Yellow fever was once common in the southeastern US, with annual outbreaks of more than 25,000 infections in New Orleans in the mid-1800s. [1] In the early 20th century, efforts to eradicate the virus that causes yellow fever were successful thanks to vaccination programs and effective control (mainly through the insecticide dichlorodiphenyltrichloroethane [DDT]) of Aedes aegypti , the mosquito that serves as a vector. Today, the virus has been largely eradicated in North America.
Elsewhere, efforts to contain yellow fever have been less successful. Despite mass vaccination campaigns in some regions, the risk for yellow fever epidemics is rising in dense urban cities in Africa and South America. [2] In an increasingly globalized society, yellow fever could easily make a comeback in North America, where A. aegypti is still present. If these mosquitoes were exposed to infected individuals, new outbreaks would be possible.
Like yellow fever, many of the circulatory and lymphatic diseases discussed in this chapter are emerging or re-emerging worldwide. Despite medical advances, diseases like malaria, Ebola, and others could become endemic in the US given the right circumstances.
Chapter Outline
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- 15.1: Arthropod-Borne Viral Diseases
- There are a number of arthropod-borne viruses, or arboviruses, that can cause human disease. Among these are several important hemorrhagic fevers transmitted by mosquitoes. We will discuss three that pose serious threats: yellow fever, chikungunya fever, and dengue fever.
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- 15.2: Parasitic Infections of the Circulatory and Lymphatic Systems
- Some protozoa and parasitic flukes are also capable of causing infections of the human circulatory system. Although these infections are rare in the US, they continue to cause widespread suffering in the developing world today. Malaria, toxoplasmosis, babesiosis, Chagas disease, leishmaniasis, and schistosomiasis are discussed in this section.
Footnotes
- Centers for Disease Control and Prevention. “The History of Yellow Fever.” http://www.cdc.gov/travel-training/local/HistoryEpidemiologyandVaccination/page27568.html
- C.L. Gardner, K.D. Ryman. “Yellow Fever: A Reemerging Threat.” Clinical Laboratory Medicine 30 no. 1 (2010):237–260.