Learn Extracted exam questions AP Physics 1 2024 Free Response
2024 Free Response
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[Figure showing a track: Point A is at the top of a curved ramp at height $6R$ above the horizontal, with a block of mass $M$ shown as a small square on the ramp near A. The ramp curves down to the horizontal level. On the horizontal section there is a small circular loop (Point B at its highest point, radius $R$) and further along, at the far right, a larger circular loop (Point C at its highest point, radius $R$) that sits on a raised platform of height $2R$ above the horizontal ground level.]
A block of mass $M$ is released from rest at Point A, a height $6R$ above the horizontal. After being released, the block slides down a track, as shown. When released from Point A, the block does not lose contact with the track at any point. Points B and C are located at the highest points of their respective circular loops, both of radius $R$. All frictional forces are negligible.
[Two energy bar chart diagrams, each with a vertical "Energy" axis (no numerical scale, gridlines shown as horizontal dashed lines) and two columns labeled $U_g$ and $K$ on the horizontal axis.]
Diagram A shows an energy bar chart that represents the gravitational potential energy $U_g$ of the block-Earth system and the kinetic energy $K$ of the block at Point A, when the block is released from rest at height $6R$. In Diagram A, the $U_g$ column is shaded up to a height representing the full scale (six gridline units above zero), and the $K$ column has zero height (block released from rest).
Diagram B has the same axes ("Energy" vertical axis with the same gridline scale, columns labeled $U_g$ and $K$) but is unshaded, to be completed by the student.
(a) Draw shaded regions in Diagram B that represent the gravitational potential energy $U_g$ and kinetic energy $K$ of the block-Earth system when the block is located at Point B, a height $2R$ above the horizontal.
- Shaded regions should start at the dashed line that represents zero energy.
- Represent any energy that is equal to zero with a distinct line on the zero-energy line.
- The relative height of each shaded region should reflect the magnitude of the respective energy consistent with the scale shown in Diagram A.
Starting with conservation of energy, derive an expression for the speed of the block at Point B. Express your answer in terms of $R$ and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.
[A single dot representing the block, drawn on the page, with blank space around it for the student to draw force vectors.]
On the following dot that represents the block, draw and label the forces (not components) that are exerted on the block at the instant the block slides through Point C. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
A student claims that $4R$ is the minimum height of Point A, such that the block can slide through Point C without losing contact with the track after the block is released from rest. Briefly explain why this claim is incorrect.
[Figure 1: A stand with a horizontal support arm (labeled "Stand") has a spring hanging vertically from the arm, with a small loop at the bottom of the spring. To the right, a set of stacked cylindrical masses of various sizes (labeled "Cylinders") sits on the table, available for hanging on the spring's loop.]
A student hangs a spring of unknown spring constant $k$ vertically by attaching one end to a stand, as shown in Figure 1. The other end of the spring has a small loop from which small cylinders can be hung. In addition to the spring, the student has access $\underline{\text{only}}$ to a variety of cylinders of unknown masses, a stopwatch, and a digital scale.
(a) Design an experimental procedure the student could use to determine the spring constant $k$ of the spring.
In the following table, list the quantities that would be measured using only the provided equipment in your experiment. Define a symbol to represent each quantity.
In the space below the table, describe the overall procedure. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table. If needed, you may include a simple diagram with your procedure.
| Quantity to Be Measured | Symbol for Quantity | Equipment for Measurement |
|---|---|---|
| Stopwatch | ||
| Digital scale | ||
Procedure (and diagram, if needed):
(b)
i. Indicate the quantities that could be plotted to produce a linear graph whose slope can be used to determine the spring constant $k$ of the spring.
Vertical axis: ________________ Horizontal axis: ________________
ii. Briefly describe how the slope of the graph would be analyzed to determine the spring constant $k$ of the spring.
[Figure 2: A force sensor is mounted on a wall on the left. A spring connects the force sensor to a cart of mass $m = 0.25\text{ kg}$ on wheels, sitting on a horizontal track. To the right of the cart is a motion detector facing the cart.]
In a different experiment, the student attaches one end of a spring to a force sensor that is attached to a wall. The other end of the spring is attached to a cart with mass $m = 0.25\text{ kg}$. The student places a motion detector to the right of the cart, as shown in Figure 2, and pulls the cart to the right a small distance so that the spring is stretched. The student releases the cart from rest, and the cart-spring system oscillates.
The following graphs show the velocity $v$ of the cart and the force $F$ exerted on the cart by the spring as functions of time $t$.
[Graph 1: velocity $v$ (m/s) vs. time $t$ (s); y-axis from $-0.4$ to $0.4$ in increments of $0.1$; x-axis from $0$ to $3.0$ s with gridlines at $0.5, 1.0, 1.5, 2.0, 2.5, 3.0$. The curve is a smooth cosine-like oscillation: starting near $v \approx 0$ at $t=0$, rising to a positive peak near $v \approx 0.3$–$0.35$ around $t \approx 0.5$ s, back down through zero near $t \approx 1.0$–$1.2$ s, down to a negative peak near $v \approx -0.3$–$-0.35$ around $t \approx 1.7$ s, back up through zero near $t \approx 2.2$–$2.4$ s, and up to another positive peak near $t \approx 2.9$–$3.0$ s. Period of oscillation is approximately $1.7$–$1.8$ s.]
[Graph 2: force $F$ (N) vs. time $t$ (s); y-axis from $-0.08$ to $0.08$ in increments of $0.02$; x-axis from $0$ to $3.0$ s with gridlines at $0.5, 1.0, 1.5, 2.0, 2.5, 3.0$. The curve is a smooth oscillation with the same period as the velocity graph, appearing approximately $90°$ out of phase with $v(t)$: starting near $F \approx 0.06$–$0.07$ (near a peak) at $t=0$, decreasing through zero around $t \approx 0.4$–$0.5$ s, reaching a negative peak near $F \approx -0.07$ around $t \approx 0.8$–$0.9$ s, back through zero near $t \approx 1.3$ s, reaching a positive peak near $t \approx 1.7$–$1.8$ s, and continuing to oscillate with the same period through $t = 3.0$ s.]
i. Using the data in the velocity-time graph, calculate the change in kinetic energy of the cart from $t = 0.5\text{ s}$ to $t = 2.0\text{ s}$. Show your steps and substitutions.
ii. Using the data in the force-time graph, estimate the change in momentum of the cart from $t = 0.5\text{ s}$ to $t = 2.5\text{ s}$. Briefly explain how you arrived at your estimation.
iii. Do the data from the velocity-time graph confirm your estimation from part (c)(ii)? Briefly explain.
[Figure 1: A vertical wall is shown on the left. Two points are marked on the wall: Point 1 near the top, and Point 2 lower down (above the hinge). A horizontal beam extends from a hinge at the base of the wall to the right, with length $L$ labeled along its underside. A string connects from Point 1 on the wall diagonally down to the right end of the beam, making angle $\theta_1$ with the beam at the point where the string meets the beam.]
The left end of a uniform beam of mass $M$ and length $L$ is attached to a wall by a hinge, as shown in Figure 1. One end of a string with negligible mass is attached to the right end of the beam. The other end of the string is attached to the wall above the hinge at Point 1. The beam remains horizontal. The hinge exerts a force on the beam of magnitude $F_H$, and the angle between the beam and the string is $\theta = \theta_1$.
[A rectangle representing the beam, drawn horizontally, blank for the student to draw and label force vectors.]
The following rectangle represents the beam in Figure 1. On the rectangle, draw and label the forces (not components) exerted on the beam. Draw each force as a distinct arrow starting on, and pointing away from, the point at which the force is exerted.
[Figure 2: The same wall, hinge, and beam of length $L$ as in Figure 1, but now the string (solid line) connects from Point 2 on the wall (lower than Point 1) to the right end of the beam, making angle $\theta_2$ with the beam. A dashed line is also shown from Point 1 to the right end of the beam, representing the string's position from Figure 1.]
(b) The string is then attached lower on the wall, at Point 2, and the beam remains horizontal, as shown in Figure 2. The angle between the beam and the string is $\theta = \theta_2$. The dashed line represents the string shown in Figure 1.
The magnitude of the tension in the string shown in Figure 1 is $F_{T1}$. The magnitude of the tension in the string shown in Figure 2 is $F_{T2}$. Indicate which of the following correctly compares $F_{T2}$ with $F_{T1}$.
____ $F_{T2} > F_{T1}$ ____ $F_{T2} < F_{T1}$ ____ $F_{T2} = F_{T1}$
Briefly justify your answer, using qualitative reasoning beyond referencing equations.
(c) Starting with Newton's second law in rotational form, derive an expression for the magnitude of the tension in the string. Express your answer in terms of $M$, $\theta$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.
(d) Is your derived equation in part (c) consistent with your justification in part (b)? Explain your reasoning.
[A blank set of axes with vertical axis labeled "Angular Speed" and horizontal axis labeled "Time", origin labeled $O$.]
(e) The string is cut, and the beam begins to rotate about the hinge with negligible friction. On the following axes, sketch the angular speed of the beam as a function of time for the time interval while the beam falls but before the beam becomes vertical.
[Figure showing a pendulum: a rigid horizontal support bracket attached to a wall/ceiling structure, from which a string hangs. The string makes angle $\theta$ with the vertical dashed line. At the end of the string is a small sphere labeled "Small Sphere", at position A. A dashed circle labeled B marks the lowest point of the swing, directly below the pivot along the vertical dashed line.]
A simple pendulum consists of a small sphere that hangs from a string with negligible mass. The top end of the string is fixed. The sphere is pulled to Point A so that the string makes a small angle $\theta$ with the vertical, as shown. The sphere is then released from rest and swings through its lowest point at Point B. The work done on the sphere by Earth between points A and B is $W_E$.
The pendulum is then taken to Planet X. The mass of Planet X is the same as the mass of Earth, but the radius of Planet X is greater than the radius of Earth. The sphere is again brought to Point A (displaced $\theta$ from the vertical), released from rest, and swings through its lowest point at Point B. The work done on the sphere by Planet X between points A and B is $W_X$.
(a) Justify why $W_X$ is less than $W_E$.
A new pendulum is made by hanging the same small sphere from a different string with negligible mass. The new string is slightly elastic, and the length of the string may increase or decrease depending on the tension applied to the string. On Earth, when the sphere is again displaced $\theta$ from the vertical and released from rest, the new pendulum oscillates with period $T_E$.
The new pendulum is then taken to a different planet, Planet Y. The radius of Planet Y is the same as the radius of Earth, but the mass of Planet Y is larger than the mass of Earth. On Planet Y, when the sphere is again displaced from the vertical and released from rest, the new pendulum oscillates with period $T_Y$.
(b) In a clear, coherent paragraph-length response that may also contain drawings, explain how $T_Y$ $\underline{\text{could be larger}}$ than $T_E$ but also $\underline{\text{could be smaller}}$ than $T_E$.
[Figure 1: Block A (mass $6\text{ kg}$) is shown on the left with three horizontal arrows/lines behind it indicating motion to the right, sitting on a horizontal surface. Block B (mass $2\text{ kg}$), smaller, sits at rest to the right of Block A on the same horizontal surface. The label "$t = 0$" is below the figure.]
At time $t = 0$, Block A slides along a horizontal surface toward Block B, which is initially at rest, as shown in Figure 1. The masses of blocks A and B are $6\text{ kg}$ and $2\text{ kg}$, respectively. The blocks collide $\underline{\text{elastically}}$ at $t = 1.0\text{ s}$, and as a result, the magnitude of the change in kinetic energy of Block B is $9\text{ J}$. All frictional forces are negligible.
(a) Determine the speed of Block B immediately after the collision.
[Figure 2: A graph of position $x$ (m) vs. time $t$ (s), with y-axis from $0$ to $6$ in increments of $1$, and x-axis from $0.0$ to $2.0$ in increments of $0.5$, gridlines at $0.0, 0.5, 1.0, 1.5, 2.0$. Three lines are drawn from $t=0$ to $t=1.0\text{ s}$: a solid line labeled "Block A" starting at $x=0$ at $t=0$ and rising linearly to about $x=2$ at $t=1.0\text{ s}$; a solid horizontal line labeled "Block B" at constant $x=2$ from $t=0$ to $t=1.0\text{ s}$ (block B at rest); and a dashed line labeled "Center of Mass" starting at about $x=0.5$ at $t=0$ and rising linearly to $x=2$ at $t=1.0\text{ s}$, in between the Block A and Block B lines.]
The graph shown in Figure 2 represents the positions $x$ of Block A, Block B, and the center of mass of the two-block system as functions of $t$ between $t = 0$ and $t = 1.0\text{ s}$.
(b) On the graph in Figure 2, draw and label three lines to represent the positions of Block A, Block B, and the center of mass of the two-block system as functions of $t$ between $t = 1.0\text{ s}$ and $t = 2.0\text{ s}$. Each line should be distinctly labeled.
(c) Consider if in the original scenario, instead of colliding elastically, the blocks collided and stuck together. Describe how the line drawn for the center of mass in part (b) would change, if at all. Briefly justify your response.