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Mock Exams AP Physics – C : Mechanics– FRQ Set 2

Question

For a particular nonlinear spring, the relationship between the magnitude of the applied force, F, and the stretch of the spring, x, is given by the equation \(F = kx^{1.5}\). How much energy is stored in the spring when is it stretched a distance \(x_1\)

(A) \(\frac{2k_1^{2.5}}{5}\)

(B) \(\frac{kx_1^{1.5}}{5}\)

(C) \(kx_1^{2.5}\)

(D) \(\frac{1}{2}kx_1^2\)

(E) \(1.5 kx_1^{0.5}\)

▶️Answer/Explanation

Ans: A

For a spring that is not linear (i.e., does not obey Hooke’s law) the energy stored is not \(\frac{1}{2} kx^2\). The magnitude of the energy stored will be equal to the magnitude of the work done to stretch the spring to \(x_1\)/ the steps to calculate the work are shown below.

Question

 In the laboratory, a block of unknown mass \(m_{1}\) is attached via a light rope and pulleys to a hanger of known mass \(m_{2}\). The pulleys are of negligible mass and turn with negligible friction. The block and hanger are released from rest in the configuration shown.

(a) Describe a procedure for measuring the acceleration of the hanger. You should use equipment usually found in a high school laboratory.

(b) Is the magnitude of the acceleration of the block greater than, less than, or equal to the magnitude of the acceleration of the hanger? Justify your answer.

Now \(m_{2}\) is varied by placing disks of different masses on the hanger. For each value of \(m_{2}\), the acceleration a of the hanger is measured. A graph is produced with \(m_{2}g\) on the vertical axis and a on the horizontal axis.

(c) Is this graph linear?
                 ___ yes                 ___ no
If yes, explain how you know and then explain how you would use a line of best fit to determine the mass \(m_{1}\).

If no, explain why not and then explain whether the graph would be concave-up or concave down.

(d) In a final experiment, the disks are removed from the hanger, and the pulleys are replaced with pulleys that do experience significant friction as they turn. Will the acceleration of the hanger in this final experiment be measured to be greater than, less than, or equal to the acceleration of the hanger in the original experiment? Justify your answer.

▶️Answer/Explanation

Ans:

(a) Many approaches. My favorite: Place a sonic motion detector beneath the hanger. Use the detector to make a velocity-time graph while the hanger is in motion. The slope of that graph is the hanger’s acceleration.

(b) Equal. Acceleration is the change in an object’s speed in some about of time. The objects are connected; if one speeds up, so does the other by the same amount. (Otherwise, they would become disconnected or the rope would go slack.)

(c) The relevant equation here comes from applying Newton’s second law to the two-object system. The net force is \(m_{2}g\) – \(m_{1}g\); the system mass is \(m_{1}\)+\(m_{2}\). Because the graph has \(m_{2}g\) on the vertical axis, solve for \(m_{2}g:m_{2}g=(m_{1}+m_{2})a+m_{1}g\). Because a appears linearly (i.e., not squared or square rooted) in the numerator, the graph is linear: yes. Identify each term with a term in the equation for a line, y = mx + b. The y variable is \(m_{2}g\). The x variable is a. This leaves the slope of the graph as (\(m_{1}\) + \(m_{2}\)). The b term is the y-intercept and is \(m_{1}g\). To get \(m_{1}\), find the vertical intercept of the best-fit line and divide by g.

(d) The newly measured acceleration would be less. Now the net force on the system isn’t just \(m_{2}g\) – \(m_{1}g\); there’s another force, the friction force, that acts opposite the direction of motion. Because the objects are released from rest, that’s also opposite the direction of acceleration. So the net force would be smaller than before. Because a = \(F_{net}/m\) and the system mass hasn’t changed, the acceleration would be smaller.

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