Learn Extracted exam questions AP Physics C: Mechanics 2019 Free Response · Set 2
2019 Free Response · Set 2
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Blocks of mass $m$ and $2m$ are connected by a light string and placed on a frictionless inclined plane that makes an angle $\theta$ with the horizontal, as shown in Figure 1 above. Another light string connecting the block of mass $m$ to a hanging sphere of mass $M$ passes over a pulley of negligible mass and negligible friction. The entire system is initially at rest and in equilibrium.
[Figure 1: An inclined plane rising to the right at angle $\theta$ from the horizontal, with a vertical wall/support on the right holding a pulley at the top. On the incline, two blocks are shown in contact, connected by a light string: a lower block labeled $2m$ and, distance $d$ up the slope from the bottom of the incline to the lower block, an upper block labeled $m$ connected to the lower block by tension $T_1$. A second string from block $m$, with tension $T_2$, runs up the incline and over the pulley at the top, then down vertically to a hanging sphere of mass $M$.]
On the dots below that represent the block of mass $m$ and the sphere of mass $M$, draw and label the forces (not components) that act on each of the objects shown. Each force must be represented by a distinct arrow starting on and pointing away from the dot.
[Two dots are shown for the student to draw free-body diagrams on: one labeled $m$, positioned on a dashed line representing the incline surface (inclined at angle $\theta$), and one labeled $M$, positioned in open space to represent the hanging sphere.]
Derive expressions for the magnitude of each of the following. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figures in part (a).
The force $T_2$ exerted on the block of mass $m$ by the string. Express your answers in terms of $m$, $\theta$, and physical constants, as appropriate.
The mass $M$ for which the system can remain in equilibrium. Express your answers in terms of $m$, $\theta$, and physical constants, as appropriate.
Now suppose that mass $M$ is large enough to descend and that the sphere reaches the floor before the blocks reach the pulley. Answer the following for the moment immediately after the sphere reaches the floor.
Does the tension $T_1$ increase, decrease to a nonzero value, decrease to zero, or stay the same?
____ Increase ____ Decrease to a nonzero value
____ Decrease to zero ____ Stay the same
Is the velocity of the block of mass $m$ up the ramp, down the ramp, or zero?
____ Up the ramp ____ Down the ramp ____ Zero
Is the acceleration of the block of mass $m$ up the ramp, down the ramp, or zero?
____ Up the ramp ____ Down the ramp ____ Zero
Consider the initial setup in Figure 1. Now suppose the surface of the incline is rough and the coefficient of static friction between the blocks and the inclined plane is $\mu_s$. Derive an expression for the minimum possible value of $M$ that will keep the blocks from moving down the incline. Express your answer in terms of $m$, $\mu_s$, $\theta$, and fundamental constants, as appropriate.
The string connecting block $m$ and the sphere of mass $M$ then breaks, and the blocks begin to move from rest down the incline. The lower block starts a distance $d$ from the bottom of the incline, as shown in Figure 1. The coefficient of kinetic friction between the blocks and the inclined plane is $\mu_k$. Derive an expression for the speed of the blocks when the lower block reaches the bottom of the incline. Express your answer in terms of $m$, $d$, $\mu_k$, $\theta$, and fundamental constants, as appropriate.
A toy rocket of mass 0.50 kg starts from rest on the ground and is launched upward, experiencing a vertical net force. The rocket's upward acceleration $a$ for the first 6 seconds is given by the equation $a = K - Lt^2$, where $K = 9.0 \text{ m/s}^2$, $L = 0.25 \text{ m/s}^4$, and $t$ is the time in seconds. At $t = 6.0$ s, the fuel is exhausted and the rocket is under the influence of gravity alone. Assume air resistance and the rocket's change in mass are negligible.
Calculate the magnitude of the net impulse exerted on the rocket from $t = 0$ to $t = 6.0$ s.
Calculate the speed of the rocket at $t = 6.0$ s.
Calculate the kinetic energy of the rocket at $t = 6.0$ s.
Calculate the change in gravitational potential energy of the rocket-Earth system from $t = 0$ to $t = 6.0$ s.
Calculate the maximum height reached by the rocket relative to its launching point.
On the axes below, assuming the upward direction to be positive, sketch a graph of the velocity $v$ of the rocket as a function of time $t$ from the time the rocket is launched to the time it returns to the ground. $T_{top}$ represents the time the rocket reaches its maximum height. Explicitly label the maxima with numerical values or algebraic expressions, as appropriate.
[Graph axes: vertical axis labeled $v$ (m/s), horizontal axis labeled $t$(s); the origin is at the intersection, with a tick mark labeled $T_{top}$ on the horizontal axis to the right of the origin. The axes are otherwise unscaled and blank for the student to sketch on.]
[Figure: A point $P$ at height $h$ above the floor, at the top of a straight incline going down and to the right, with a small circle labeled $m, r$ at $P$ representing the rolling sphere. The incline descends to a circular vertical loop of radius $R$ (labeled with a radius line from the center to the edge of the loop), with point $A$ marked at the top of the loop. After the loop, the track continues to the right and up as a straight incline. A vertical dashed line with arrows shows the height $h$ measured from point $P$ down to the level of the bottom of the loop/floor. Note: Figure not drawn to scale.]
The rotational inertia of a rolling object may be written in terms of its mass $m$ and radius $r$ as $I = bmr^2$, where $b$ is a numerical value based on the distribution of mass within the rolling object. Students wish to conduct an experiment to determine the value of $b$ for a partially hollowed sphere. The students use a looped track of radius $R \gg r$, as shown in the figure above. The sphere is released from rest a height $h$ above the floor and rolls around the loop.
Derive an expression for the minimum speed of the sphere's center of mass that will allow the sphere to just pass point $A$ without losing contact with the track. Express your answer in terms of $b$, $m$, $R$, and fundamental constants, as appropriate.
Suppose the sphere is released from rest at some point $P$ and rolls without slipping. Derive an equation for the minimum release height $h$ that will allow the sphere to pass point $A$ without losing contact with the track. Express your answer in terms of $b$, $m$, $R$, and fundamental constants, as appropriate.
The students perform an experiment by determining the minimum release height $h$ for various other objects of radius $r$ and known values of $b$. They collect the following data.
| Object | $b$ | $h$ (m) |
|---|---|---|
| Solid sphere | 0.40 | 1.08 |
| Hollow sphere | 0.67 | 1.13 |
| Solid cylinder | 0.50 | 1.10 |
| Hollow cylinder | 1.0 | 1.20 |
On the grid below, plot the release height $h$ as a function of $b$. Clearly scale and label all axes, including units, if appropriate. Draw a straight line that best represents the data.
[Blank grid with dashed gridlines for plotting $h$ versus $b$; no axes labels or scale marked, to be filled in by the student.]
The students repeat the experiment with the partially hollowed sphere and determine the minimum release height to be 1.16 m. Using the straight line from part (c), determine the value of $b$ for the partially hollowed sphere.
Calculate $R$, the radius of the loop.
In part (b), the radius $r$ of the rolling sphere was assumed to be much smaller than the radius $R$ of the loop. If the radius $r$ of the rolling sphere was not negligible, would the value of the minimum release height $h$ be greater, less, or the same?
____ Greater ____ Less ____ The same
Justify your answer.