2024 AP Physics C: Mechanics Free-Response Questions

Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this booklet in the spaces provided after each part. Figure 1: Block A, of mass m, is attached to an ideal spring on a horizontal surface. Block B, of mass 3m, is attached to a string of length ℓ. The horizontal axis is labeled x, with positions x0, x1, x2, and x3. Block A initially compresses the spring a distance xc. A region of length D has coefficient of kinetic friction μ. The figure also labels the spring constant k, the string length ℓ, and the point x=0. Note: Figure not drawn to scale. 1. Block A and Block B of masses m and 3m, respectively, are arranged in a setup consisting of an ideal spring with spring constant k and a horizontal surface. Friction between the surface and the blocks is negligible except in a region of length D, where the coefficient of kinetic friction between Block A and the surface is μ. Block B is attached to a string of length ℓ and negligible mass, as shown in Figure 1. Block A is held against the spring, compressing the spring a distance xc. At time t = 0, Block A is located at position x = x0 and is released from rest. After the block is released, the following occurs. • At time t = t1, Block A is at x = x1 after traveling a distance xc. Block A moves with speed v, and the spring is at its equilibrium position. • At time t = t2, the left side of Block A is at x = x2 after passing through a distance D across the region with nonnegligible friction. • At time t = t3, Block A is at x = x3 and Block A collides with and sticks to Block B. (a) For parts (a)(i) and (a)(ii), express your answer in terms of m, k, D, μ, xc, and physical constants, as appropriate. i. Derive an expression for the speed v of Block A at time t1. ii. Derive an expression for the speed vA,B of the two-block system immediately after the collision at time t3. (b) i. On the following axes, sketch a graph of the kinetic energy K of Block A as a function of time t from time t = 0 to time t3. The axes show K versus t, with marked times 0, t1, t2, and t3. ii. Use principles of work and energy to justify the graph drawn in part (b)(i) for the time interval t = 0 to t = t1. Explicitly reference features of the shape of the graph you drew in part (b)(i). After the collision, the two-block system instantaneously comes to rest at time t4, which occurs when the string makes a small angle θmax with the vertical, as shown in Figure 2. For times t > t4, the system oscillates with frequency fℓ. The support holding the string is raised, and the procedure is then repeated using a new string of length 2ℓ. (c) Indicate how the new frequency of oscillation f2ℓ of the system on the new string of length 2ℓ will compare to the frequency of oscillation fℓ from the original procedure. _____ f2ℓ > fℓ _____ f2ℓ < fℓ _____ f2ℓ = fℓ Briefly justify your answer.

2. A student drops a cylinder of mass m from rest. The air exerts a drag force of magnitude F_drag on the cylinder, as shown in Figure 1. The student models the magnitude of the drag force as F_drag = bv^2, where v is the speed of the cylinder and b is a positive constant with appropriate units. [Figure 1: A cylinder with an upward drag force F_drag shown.] (a) Derive, but do NOT solve, a differential equation that could be used to determine the speed v of the cylinder as a function of time t. Express your answer in terms of given quantities and physical constants, as appropriate. (b) The student correctly sketches the speed v of the cylinder as a function of time t, as shown in Figure 2. [Figure 2: A graph of v versus t. The curve starts at the origin, increases rapidly, and approaches a horizontal maximum-speed value labeled v_max.] i. Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which F_drag on the cylinder is equal to the magnitude of the weight of the cylinder. Label this time as t_1 on the time axis. ii. Justify the location of t_1. Explicitly reference appropriate features of the sketch in Figure 2. (c) Rather than dropping the cylinder from rest, the student throws the cylinder upward with a nonzero initial speed. The cylinder is in the same orientation as when the cylinder was previously dropped. The student allows the cylinder to fall toward the ground. Indicate whether the magnitude of the cylinder’s maximum downward speed after being thrown upward would be greater than, less than, or equal to the maximum speed v_max in Figure 2. _____ Greater than _____ Less than _____ Equal to Briefly justify your answer. (d) The student conducts an experiment to better understand the relationship between maximum speed v_max and mass. The student collects data to determine the maximum speed for cylinders dropped from rest, each with the same physical size and shape but a different mass m. The student then graphs v_max^2 as a function of mass. [Graph: vertical axis v_max^2 (m^2/s^2), horizontal axis m (kg), with data points plotted at approximately (0.1, 2.3), (0.2, 3.6), (0.3, 5.2), (0.4, 7.5), and (0.5, 8.5).] i. Draw the best-fit line for the data. ii. Use the best-fit line to calculate an experimental value for b. A student claims that the magnitude of the maximum speed of a cylinder dropped from rest depends on the length of the cylinder. The student designs an experiment to collect data that can be used to provide evidence to support the claim. The student drops cylinders with the orientation shown in Figure 3. [Figure 3: A cylinder with the upward drag force F_drag shown and a double-headed arrow underneath labeled Length.] (e) The student has access to but does not have to use all of the following equipment. • Cylinder Set 1: cylinders of the same known length with different known masses • Cylinder Set 2: cylinders of the same known mass with different known lengths • A motion detector that can measure velocity as a function of time i. Indicate two quantities that when graphed could be used to determine whether the length of the cylinder affects the maximum speed. Vertical axis: ____________ Horizontal axis: ____________ ii. Briefly describe how the quantities graphed could be used to determine the relationship between cylinder length and maximum speed.

3. A uniform rod of length L and mass m is attached to a pivot on a vertical pole, as shown in Figure 1. There is negligible friction between the rod and the pivot. A horizontal string connects Point Q on the rod to the pole. The rod makes an angle θ with the pole. A block of mass 3m hangs from the rod at Point P. The center of mass of the rod is located at Point C. Figure 1 shows the pivot, the rod of length L, Points P, C, and Q, the horizontal string attached at Q, the angle θ between the rod and the vertical pole, and the block of mass 3m hanging at P. Note: Figure not drawn to scale. (a) On the following representation of the rod, draw and label the forces (not components) that are exerted on the rod. Each force must be represented by a distinct arrow that starts on and points away from the point at which the force is exerted on the rod. The answer diagram shows the rod with the pivot and Points P, C, and Q labeled. (b) In Figure 1, Point P is located 3/8 L from the pivot and Point Q is located 6/8 L from the pivot. Derive an equation for the tension F_T in the horizontal string in terms of L, m, θ, and physical constants, as appropriate. Figure 2 shows the original string replaced with a longer string that connects Point Q to a higher location on the vertical pole. The angle θ remains the same. Note: Figure not drawn to scale. (c) The original string is replaced with a longer string that connects Point Q to a higher location on the vertical pole, as shown in Figure 2. The angle θ remains the same. How does the new tension F_T,new compare with the original tension F_T from part (b)? Justify your reasoning. Figure 3 shows a nonuniform rod attached to a pivot. The rod extends along the x-axis from x = 0 at the pivot to x = 1.2 m, with a string attached near the right end. Note: Figure not drawn to scale. (d) A nonuniform rod is now attached to the pivot, as shown in Figure 3. There is negligible friction between the nonuniform rod and the pivot. The rod has a length of 1.2 m and a linear mass density λ(x) = A + Bx, where x is the distance from the pivot, A = 6.0 kg/m, and B = 10.0 kg/m². i. Calculate the mass of the rod. ii. Calculate the rotational inertia of the rod about the pivot.