Learn Extracted exam questions AP Physics C: Mechanics 2023 Free Response · Set 1
2023 Free Response · Set 1
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[Figure 1: A block on an inclined ramp on the left, connected via the horizontal surface to Spring P mounted against a wall on the right. The spring is compressed a distance $x$ from its natural length, with $x=0$ marked at the wall-side end of the spring where it is uncompressed.]
A block sitting on a horizontal surface is pushed against a spring, Spring P, that is attached to a wall, compressing the spring a distance $x$, as shown in Figure 1. The block is then released from rest. The block slides along the horizontal surface and up a ramp, reaching a maximum height $h_{\text{max,P}}$. Frictional forces between the block and all surfaces are negligible.
A student compares $h_{\text{max,P}}$ for Spring P with the different height $h_{\text{max,Q}}$ achieved with a different spring, Spring Q. Each spring exerts a force of magnitude $F$ on the block that varies as a function of the distance $x$ the spring is compressed, as graphed in Figure 2. For Spring P, $F_P(x) = kx$, where $k = 100.0\ \text{N/m}$, and for Spring Q, $F_Q(x) = Cx^{1/2}$, where $C = 20.0\ \text{N/m}^{1/2}$.
[Figure 2: Graph of $F$ (N) on the y-axis from 0.0 to 10.0, versus $x$ (m) on the x-axis from 0.000 to 0.100, in increments of 0.020. Two curves are plotted: a solid line labeled "Spring P" that is straight, passing through the origin and rising linearly (consistent with $F_P(x)=kx$, reaching about 10.0 N at $x=0.100$ m); a dashed curve labeled "Spring Q" that rises steeply at first and then levels off (consistent with $F_Q(x) = Cx^{1/2}$, reaching about 6.3 N at $x=0.100$ m).]
For the experiment, the block is pushed against one of the springs, compressing the spring a distance $x = 0.010\ \text{m}$. The block is then released from rest. In Trial 1, Spring P is used, and in Trial 2, Spring Q is used. On the following representations of the block in Trials 1 and 2, draw and label the forces (non-components) that are exerted on the block at the instant the block is released. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. The lengths of the horizontal vectors should represent the relative magnitude of the horizontal forces and the lengths of the vertical vectors should represent the relative magnitude of the vertical forces.
[Two grid diagrams are shown, each with a dot at the center representing the block: "Trial 1 (Spring P)" on the left and "Trial 2 (Spring Q)" on the right. Each grid is for the student to draw labeled force vectors.]
What feature(s) of the graph in Figure 2 could be used to estimate the work done on the block by each spring as each spring is compressed?
There is one compression distance $x_0$ for which the maximum height $h_{\text{max}}$ reached by the block is the same regardless of which spring, Spring P or Spring Q, is used. Predict whether the value of $x_0$ is greater than, less than, or equal to $0.040\ \text{m}$. Use the graph in Figure 2 to justify your answer.
On the axes provided, for both Spring P and Spring Q, sketch a graph of the maximum height $h_{\text{max}}$ reached by the block as a function of the distance $x$ each spring is compressed for values of $x$ ranging from 0 to 0.100 m. Clearly label the curve for Spring P and Spring Q.
[Blank axes provided: y-axis labeled $h_{\text{max}}$ (m), x-axis labeled $x$ (m) from 0.000 to 0.100 in increments of 0.020, for the student to sketch two curves.]
[Figure 3: Block A rests at the top of an inclined ramp of height $H$, connected by a string over the top of the ramp; Block B rests on the horizontal surface at the bottom of the ramp, connected to Spring Q which is attached to a wall, with $x=0$ marked at the natural length position of the spring.]
Spring Q is attached to the wall and at equilibrium, as shown in Figure 3. Block A has a mass of 0.120 kg and Block B has a mass of 0.070 kg. Block A is held at rest at the top of the ramp on the horizontal surface. The change in vertical height of Block A is $H = 0.75\ \text{m}$. The student releases Block A, and it moves down the ramp and collides with Block B. After the collision, the blocks stick together and move to the right, compressing the spring. Frictional forces between the blocks and all surfaces are negligible.
Calculate the velocity of the two-block system immediately after the collision between Blocks A and B.
Calculate the maximum compression of the spring.
Spring Q where $F_Q(x) = Cx^{1/2}$ is replaced by a different nonlinear spring, Spring R, and the procedure described in part (c) is repeated. For Spring R, $F_R(x) = Dx^{1/2}$. The maximum compression of Spring R is greater than the maximum compression of Spring Q. Which of the following correctly compares the constants $C$ and $D$?
_____ $C < D$ _____ $C > D$ _____ $C = D$
Briefly justify your answer.
[Figure 1: A torsional pendulum with a single uniform disk ($N=1$) suspended from a light wire, showing an angular displacement $\theta_0$ from the untwisted position. Figure 2: The same torsional pendulum with a second identical disk attached below the first ($N=2$), also displaced by angle $\theta_0$.]
A student makes a torsional pendulum by suspending a uniform disk of mass $M$ and radius $R$ from a light wire with torsion constant $\kappa$ that is attached to the center of the disk as shown in Figure 1. The rotational inertia of the disk is given by $I = \dfrac{1}{2}MR^2$. The student conducts an investigation to determine the relationship between the period of oscillation $T$ of the torsional pendulum and the number $N$ of identical disks that are suspended from the wire.
The student starts with a single disk. Holding the disk at a small initial angular displacement $\theta_0$ from the untwisted position, the student releases the disk from rest and the pendulum oscillates. The student records the period of oscillation for a single disk. An additional identical disk is attached, as shown in Figure 2, and the procedure is repeated for $N = 2$ disks. This procedure is repeated through $N = 10$ identical disks. Assume the disks move together as one system.
Using $T = 2\pi\sqrt{\dfrac{I}{\kappa}}$, derive an expression for $T$ as a function of $N$. Express your answer in terms of $M$, $R$, $\kappa$, $N$, and physical constants, as appropriate.
The potential energy stored in the torsional pendulum when the disks are displaced is $U = \dfrac{1}{2}\kappa(\Delta\theta)^2$. On the following axes, sketch a graph of the maximum kinetic energy $K_{\text{max}}$ of the torsional pendulum as a function of $N$ for $N \ge 1$.
[Blank axes provided: y-axis labeled $K_{\text{max}}$, x-axis labeled $N$, for the student to sketch a curve.]
[Figure: Graph of $T$ (s) on the y-axis from 0 to 0.8 in increments of 0.1, versus $\sqrt{N}$ on the x-axis from 0 to 2.5 in increments of 0.5. Scattered data points are plotted showing an approximately linear increasing trend, roughly from $(\sqrt{N}\approx0.7, T\approx0.25)$ up to $(\sqrt{N}\approx2.3, T\approx0.65)$, for the student to draw a best-fit line through.]
The student plots the data for $T$ as a function of $\sqrt{N}$, as shown.
Draw the best-fit line for the data.
The student previously determined that the radius of a disk is $R = 0.2\ \text{m}$ and found that $\kappa = 1.6\ \text{N}\cdot\text{m}$. Using the graph, calculate the mass $M$ of a single disk.
The student finds that the value given by the manufacturer for the mass of the disk is less than the value determined experimentally in part (c)(ii). Determine a single source of experimental error that could result in the observed difference in the value of $M$. Justify your answer.
The student repeats the experiment, but now the disks have a density that varies as a function of the radius of the disk according to $\rho = 0.3r$. Would the slope of the best-fit line for this new data be greater than, less than, or the same as the slope of the best-fit line in part (c)(i)?
_____ greater than _____ less than _____ the same as
Justify your answer.
When $N = 1$, the maximum angular speed of the torsional pendulum with a uniform disk is found to be $\omega_{\text{U}}$. When $N = 1$, the maximum angular speed of the torsional pendulum with a nonuniform disk is $\omega_{\text{non-U}}$. Which of the following correctly compares $\omega_{\text{U}}$ and $\omega_{\text{non-U}}$?
_____ $\omega_{\text{U}} > \omega_{\text{non-U}}$ _____ $\omega_{\text{U}} < \omega_{\text{non-U}}$ _____ $\omega_{\text{U}} = \omega_{\text{non-U}}$
Briefly justify your answer.
[Figure 1: A horizontal surface with Point B on the left and Point A near the middle marked by a black dot (the sphere's initial position at rest). A rod is shown above Point A, extending upward and connected at a pivot to a small block/weight at the top, with the rod initially vertical making a $90°$ angle at the pivot. A dashed curved arrow shows the rod swinging down from vertical to horizontal, striking the sphere at Point A. Note: Figure not drawn to scale.]
A system consists of a small sphere of mass $m$ and radius $R$ at rest on a horizontal surface and a uniform rod of mass $M = 2m$ and length $\ell$ attached at one end to a pivot with negligible friction, where $R \ll \ell$. There is negligible friction between the surface and the sphere to the right of Point A and nonnegligible friction to the left of Point A. The rod is held horizontally as shown in Figure 1, then is released from rest. The total rotational inertia of the rod about the pivot is $\dfrac{1}{3}M\ell^2$ and the rotational inertia of the sphere about its center is $\dfrac{2}{5}mR^2$. After the rod is released, the rod swings down and strikes the sphere head-on. As a result of this collision, the rod is stopped, and the ball initially slides without rotating to the left across the horizontal surface.
Derive an expression for the angular speed of the rod just before striking the sphere in terms of the length $\ell$ and physical constants as appropriate.
Derive an expression for the linear speed $v_0$ of the sphere immediately after colliding with the rod in terms of the length $\ell$ and physical constants as appropriate.
After sliding a short distance, at time $t = 0$ the sphere encounters a region of the horizontal surface with a coefficient of kinetic friction $\mu$, beginning at Point A as indicated in Figure 1. The sphere begins rotating while sliding and eventually begins rolling without sliding at Point B, also as indicated.
In the following diagram, which represents the sphere while the sphere is traveling between Points A and B, draw and label the forces (not components) that act on the sphere. Each force must be represented by a distinct arrow starting on, and pointing away from, the point of application on the sphere.
[A circle is shown representing the sphere, for the student to draw and label force vectors starting on the circle.]
Derive an expression for each of the following as the sphere is rotating and sliding between points A and B in terms of $v_0$, $\mu$, $R$, $t$, and physical constants as appropriate.
The linear velocity $v$ of the center of mass of the sphere as a function of time $t$
The angular velocity $\omega$ of the sphere as a function of time $t$
Derive an expression for the time it takes the sphere to travel from Point A to Point B in terms of $v_0$, $\mu$, and physical constants as appropriate.
Derive an expression for the linear velocity of the sphere upon reaching Point B in terms of $v_0$.