Learn Extracted exam questions AP Physics C: Mechanics 2024 Free Response · Set 2
2024 Free Response · Set 2
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Blocks A and B of masses $2m$ and $m$, respectively, are arranged in a setup consisting of a ramp that makes an angle $\theta$ with a smooth horizontal table and an ideal spring of spring constant $k$ fixed to a wall, as shown. Block A is held at rest a distance $D$ up the ramp, and Block B is at rest on the horizontal table. The coefficient of kinetic friction between Block A and the rough ramp is $\mu$ in the region of length $D$, and there is negligible friction between the blocks and the smooth table.
[Figure: A ramp inclined at angle $\theta$ to a horizontal table. Block A (mass $2m$) sits on the ramp at a distance $D$ up the incline, with the region of length $D$ on the ramp labeled with friction coefficient $\mu$. The ramp meets the table at the origin $0$. On the horizontal table, Block B (mass $m$) rests at position $x_1$. Further right along the table, a spring of constant $k$ is fixed to a wall, with its uncompressed (equilibrium) position at $x_3$; a distance $x_c$ is marked from $x_3$ back toward the wall/spring, near the wall. The horizontal axis is labeled with positions $0$, $x_1$, $x_3$ increasing to the right toward the wall.]
Note: Figure not drawn to scale.
Continue your response to QUESTION 1 on this page.
At time $t=0$, Block A is located at horizontal position $x=0$ and is released from rest. After the block is released, the following occurs.
- At time $t=t_1$, Block A has traveled a distance $D$ down the ramp, has transitioned to the table, and is moving with speed $v$ at $x=x_1$.
- At time $t=t_2$, Block A is at $x=x_2$ when it collides with and sticks to Block B.
- At time $t=t_3$, the combined blocks A and B are at $x=x_3$ when they collide with and stick to the spring in its equilibrium position.
- At time $t=t_4$, the combined blocks A and B are instantaneously at rest and the spring is compressed a distance $x_c$ from its equilibrium position.
For parts (a)(i) and (a)(ii), express your answer in terms of $m$, $\theta$, $D$, $\mu$, $x_c$, and physical constants, as appropriate.
Derive an expression for the speed $v$ of Block A at time $t_1$.
Derive an expression for the spring constant $k$ of the spring.
[Figure repeated: the same ramp-table-spring setup with Block A (mass $2m$) on the ramp, Block B (mass $m$) on the table, and the spring of constant $k$ near the wall; axis positions $0$, $x_1$, $x_2$, $x_3$ marked left to right, with $x_c$ marked near the spring. Note: Figure not drawn to scale.]
On the following axes, sketch a graph of the magnitude of the momentum $p_A$ of Block A as a function of time $t$ from $t=0$ to $t_4$.
[Graph: vertical axis labeled $p_A$ (unlabeled scale), horizontal axis labeled $t$ with tick marks at $0$, $t_1$, $t_2$, $t_3$, $t_4$ (unlabeled scale/blank grid for the student to draw on).]
Use principles of forces to justify the graph drawn in part (b)(i) for the time interval $t=t_3$ to $t=t_4$. Explicitly reference features of the shape of the graph you drew in part (b)(i).
For times $t>t_4$, the two-block-spring system oscillates with period $T_O$. The procedure is then repeated using a new ramp, where there is negligible friction between Block A and the ramp.
Indicate how the new period of oscillation $T_N$ in the procedure that uses the new ramp compares with the period of oscillation $T_O$ from the original procedure.
____ $T_N > T_O$ ____ $T_N < T_O$ ____ $T_N = T_O$
Briefly justify your answer.
[Figure 1: A sphere of diameter $D$ falls downward through the air; arrows above the sphere point downward representing airflow/motion, and an arrow labeled $F_{drag}$ points upward from the top of the sphere, opposing the motion. The sphere is labeled "Sphere" with its diameter $D$ marked across it.]
A student drops a sphere of mass $m$ from rest. The air exerts a drag force of magnitude $F_{drag}$ on the sphere, as shown in Figure 1. The student models the magnitude of the drag force as $F_{drag} = bv$, where $v$ is the speed of the sphere and $b$ is a positive constant with appropriate units.
Derive, but do NOT solve, a differential equation that could be used to determine the speed $v$ of the sphere as a function of time $t$. Express your answer in terms of given quantities and physical constants, as appropriate.
[Figure 2: A graph of drag force $F_{drag}$ (vertical axis) versus time $t$ (horizontal axis), both axes starting at $0$. The curve starts at the origin, rises steeply, then curves over and levels off to a constant (horizontal) value at large $t$.]
The student sketches the drag force $F_{drag}$ exerted on the sphere as a function of time $t$, as shown in Figure 2.
Draw a vertical line on the sketch in Figure 2 to indicate the earliest time at which $F_{drag}$ is equal to the magnitude of the weight of the sphere, which occurs when the sphere reaches terminal speed. Label this time as $t_T$ on the time axis.
Justify the location of $t_T$. Explicitly reference appropriate features of the sketch in Figure 2.
Suppose the student throws the same sphere downward with a nonzero initial speed. The magnitude of the new drag force at terminal speed after being thrown downward is $F_{new}$.
Indicate whether $F_{new}$ would be greater than, less than, or equal to the magnitude of $F_{drag}$ at terminal speed represented in Figure 2.
____ Greater than ____ Less than ____ Equal to
Briefly justify your answer.
The student conducts an experiment to better understand the relationship between $F_{drag}$ and $v$. The student makes measurements to calculate and graph the magnitude of $F_{drag}$ as a function of $v$ for the falling sphere.
[Graph: scatter plot of $F_{drag}$ (N) on the vertical axis, from $0$ to $20$ in increments of $5$, versus $v$ (m/s) on the horizontal axis, from $0$ to $25$ in increments of $5$, on a fine grid. Data points (approximate positions read from the grid): $(3, 3)$, $(6, 6)$, $(9, 6.5)$, $(12, 9.5)$, $(15, 13.5)$, $(18, 17.5)$, $(20, 18)$, $(21, 18.5)$. The points show a roughly linear increasing trend with some scatter.]
Draw the best-fit line for the data.
Use the best-fit line to calculate an experimental value for $b$.
A student claims that the terminal speed $v_T$ of the sphere depends on the diameter $D$ of the sphere. The student designs an experiment to collect data that can be used to provide evidence to support the claim.
The student has access to but does not have to use all of the following equipment.
- Sphere Set 1: spheres of the same known mass with different known diameters
- Sphere Set 2: spheres of the same known diameter with different known masses
- A motion detector that can measure velocity as a function of time
Indicate two quantities that when graphed could be used to determine whether the diameter of the sphere affects the terminal speed.
Vertical axis: ________________ Horizontal axis: ________________
Briefly describe how the quantities graphed could be used to determine the relationship between sphere diameter and terminal speed.
[Figure 1: A uniform disk mounted on a horizontal axle that passes through a vertical pole, with the axle at the disk's center. A string runs horizontally from the pole to point A at the edge of the disk near the top; the string makes angle $\theta$ with the line between point A and the axle. A lump of clay of mass $m_c$ is attached to the edge of the disk at point A. Another lump of clay of mass $m_d$ hangs from a string on the lower-left edge of the disk. The disk radius is labeled $R$, measured from the axle to the edge. The disk sits on a vertical pole mounted on a circular base.]
Note: Figure not drawn to scale.
A uniform disk of radius $R$ and mass $m_d$ is attached to a vertical pole by a horizontal axle that passes through the center of the disk. Friction between the axle and the disk is negligible. A lump of clay of mass $m_c$ is attached to the edge of the disk at Point A. The size of the lump of clay is small compared with the radius of the disk. A horizontal string is connected from the pole to the edge of the disk at Point A. The string makes an angle $\theta$ with the line between Point A and the axle, as shown in Figure 1.
On the following representation of the clay-disk system, draw and label the external forces (not components) exerted on the system. 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 system.
[Figure: A shaded disk viewed face-on, with a dot at the center marking the axle, and point A marked on the upper-right edge of the disk (blank, for the student to draw force arrows on).]
Derive an expression for the tension $F_T$ in the string when the clay is at Point A, as shown in Figure 1, in terms of $R$, $m_d$, $m_c$, $\theta$, and physical constants, as appropriate.
[Figure 2: The same disk-axle-pole setup, now with point A marked on the upper edge of the disk and point B marked at the right edge of the disk, horizontally in line with the axle; a dashed horizontal line connects the axle to point B. Note: Figure not drawn to scale.]
The string remains connected to the edge of the disk at Point A. The clay is moved to Point B, which is horizontally in line with the axle, as shown in Figure 2. How does the new tension $F_{T,new}$ compare with tension $F_T$ from part (b)? Justify your reasoning.
[Figure 3: A nonuniform disk (shown with radial shading/stippling denoting varying density) mounted on the same axle-pole setup, labeled "Nonuniform Disk, $\rho(r) = \beta r$". Point A is marked on the upper edge of the disk, and point B is marked at the right edge of the disk, horizontally in line with the axle (dashed line from axle to B). The disk radius is labeled $R$. Note: Figure not drawn to scale.]
A nonuniform disk is now attached to the axle. The lump of clay is attached to the disk at Point B, as shown in Figure 3. The clay has mass $m_c = 0.60\text{ kg}$ and the disk has a radius $R = 0.30\text{ m}$. The mass density of the disk varies radially and can be modeled by $\rho(r) = \beta r$, where $r$ is the radial distance from the axle and $\beta = 4.0\text{ kg/m}^3$.
Calculate the rotational inertia of the disk about the axle.
The string connecting the disk to the pole is cut. Calculate the magnitude of the initial angular acceleration of the clay-disk system.