Learn Extracted exam questions AP Physics C: Mechanics 2021 Free Response · Set 2
2021 Free Response · Set 2
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Students design an experiment using blocks of adjustable mass to investigate friction using the setup shown. Block 1 of initial mass 0.44 kg is placed on a rough horizontal surface and connected by a string to block 2 of initial mass 0.20 kg. The string extends over a pulley that has negligible mass and friction.
[Figure: Setup diagram. Motion Detector 1 is at the left, facing right toward Block 1 (0.44 kg), which sits on a horizontal rough surface. A string from Block 1 runs right over a frictionless, massless pulley at the edge of the surface, then down to Block 2 (0.20 kg), which hangs vertically. Motion Detector 2 is below Block 2, facing up toward it.]
Calculate the minimum value of the coefficient of static friction $\mu_s$ that would keep the two-block system at rest.
The coefficient of friction is such that when block 2 is released from rest, block 1 travels across the surface. The acceleration $a$ of each block is recorded with motion detectors 1 and 2, as shown in the figure. The data for the motion detectors as functions of time $t$ are shown on the graphs. For each motion detector, the positive direction is away from the detector.
[Two graphs side by side, each with vertical axis $a$ (m/s$^2$) ranging from $-2$ to $2$, and horizontal axis $t$ (s) ranging from $0.0$ to $1.5$, gridlines at intervals of $0.5$ s and $1$ m/s$^2$. Left graph "Motion Detector 1": data points scattered near $a \approx 2\ \text{m/s}^2$ for all shown $t$ from $0.0$ to $1.5$ s (roughly constant, slightly above the $a=2$ gridline). Right graph "Motion Detector 2": data points scattered near $a \approx 2\ \text{m/s}^2$ for all shown $t$ from $0.0$ to $1.5$ s (roughly constant, similar to the left graph).]
On the dots below, which represent the blocks, draw and label the forces (not components) that act on each block. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
[Two labelled dots are provided for the student to draw on: "Block 1" and "Block 2", each shown as a single black dot with blank space around it for force arrows.]
Calculate the coefficient of kinetic friction $\mu_k$ between block 1 and the table.
Careful measurements determine that the coefficient of kinetic friction is larger than the value calculated in part (c). Does the following explanation sufficiently account for the observed discrepancy?
"The horizontal table was not perfectly level before the experiment was conducted. The observed difference in the angle accounts for the difference in the expected and calculated values of $\mu_k$."
_____Yes _____No
Justify your answer.
The experiment is moved to a surface with negligible friction and run for eight trials. In each trial, the students vary the masses $m_1$ and $m_2$ of blocks 1 and 2, respectively, while keeping the total mass $(m_1 + m_2) = 0.64\ \text{kg}$ constant. The data for the acceleration $a$ of block 1 as a function of $m_2$ are shown on the graph below.
[Graph: vertical axis $a$ (m/s$^2$) with gridlines at $2.0$, $4.0$, $6.0$; horizontal axis $m_2$ (kg) with gridlines at $0.20$, $0.40$. Eight data points show an increasing, slightly concave/roughly linear trend: starting near $(0.05, 1.3)$, rising through approximately $(0.10, 2.0)$, $(0.15, 3.0)$, $(0.20, 3.3)$, $(0.25, 4.2)$, $(0.30, 4.9)$, $(0.35, 5.3)$, up to about $(0.40, 5.6)$.]
Draw a best-fit line for the data points.
Using the straight line, calculate an experimental value for the acceleration due to gravity $g$.
The students lift the left end of the surface so that the surface is inclined at an angle to the horizontal, and the experiment for $m_2 = 0.20\ \text{kg}$ is repeated. Would the acceleration of the system be greater than, less than, or equal to the acceleration of the system in the original experiment?
_____Greater than _____Less than _____Equal to
Justify your claim.
[Figure: "Object A" — a long, thin, uniform rod of mass $M$ and total length $2L$, drawn horizontally. A pivot is shown at the left end (marked "Pivot" with a bracket symbol), and the rod extends to the right; a bracket below the rod labels its full length as $2L$, with $M$ labelling the mass above the rod.]
Object A is a long, thin, uniform rod of mass $M$ and length $2L$ that is free to rotate about a pivot of negligible friction at its left end, as shown above.
Using integral calculus, derive an expression to show that the rotational inertia $I_A$ of object A about the pivot is given by $\dfrac{4}{3}ML^2$.
[Figure: "Object B" — an L-shaped (right-angle) object formed of two rods, shown on an $x$-$y$ coordinate system with origin $O$ at the bottom-left. A pivot is marked at the top-left corner (labelled "Pivot", on the $y$-axis). A horizontal rod of length $L$ extends right from the pivot along the top (labelled "$L$"), and a vertical rod of length $L$ extends down from the pivot to $O$ along the $y$-axis (labelled "$L$"). The $x$-axis points right from $O$, the $y$-axis points up from $O$ through the pivot.]
Object B of total mass $M$ is formed by attaching two thin, uniform, identical rods of length $L$ at a right angle to each other. Object B is held in place, as shown above. Express your answers in part (b) in terms of $L$.
Determine the following for the given coordinate system shown in the figure.
The $x$-coordinate of the center of mass of object B
The $y$-coordinate of the center of mass of object B
Object B has a rotational inertia of $I_B$ about its pivot.
Is the value of $I_B$ greater than, less than, or equal to $I_A$?
_____Greater than _____Less than _____Equal to
Justify your answer.
Object B is released from rest and begins to rotate about its pivot.
On the axes below, sketch graphs of the magnitude of the angular acceleration $\alpha$ and the angular speed $\omega$ of object B as functions of time $t$ from the time it is released to the time its center of mass reaches its lowest point.
[Two blank sets of axes are provided: left axis labelled $\alpha$ (vertical) vs. $t$ (horizontal); right axis labelled $\omega$ (vertical) vs. $t$ (horizontal). Both axes are otherwise unmarked (no gridlines or scale), for the student to sketch curves.]
[Figure: The L-shaped object B is shown rotated partway from its initial horizontal/vertical position, tilted through an angle $\theta$ (marked at the top between a dashed horizontal reference line and the now-tilted top rod), with a curved arrow indicating the direction of rotation (downward/clockwise) at the lower end of the shape.]
While object B rotates from the horizontal position down through the angle $\theta$ shown above, is the magnitude of its angular acceleration increasing, decreasing, or not changing?
_____Increasing _____Decreasing _____Not changing
Justify your answer.
[Figure: Object B is shown in a new orientation — the vertical rod of length $L$ now lies along the bottom (horizontal), and the horizontal rod of length $L$ extends straight up from its right end (vertical), forming an upside-down L / right-angle shape resting on the ground.]
Object B rotates through the position shown above.
Derive an expression for the angular speed of object B when it is in the position shown above. Express your answer in terms of $M$, $L$, $I_B$, and physical constants, as appropriate.
[Figure: A block of mass $m$ sits on top of a vertical spring (spring constant $k$) which is compressed a distance $\Delta x$ from its natural length, at the bottom of a vertical track. The track curves upward and to the right in a smooth arc, reaching a highest point labelled $A$ at height $3R$ above the block's release point, where the track has constant radius of curvature $R$ (shown with a radius arrow labelled $R$ from a center point to point $A$). The track continues curving over the top and down to point $B$, at the same height $3R$, where the track becomes horizontal (radius $R$ again, shown with a radius arrow labelled $R$ from a center point to $B$) and the block exits horizontally moving to the right and slightly upward along a dashed projectile trajectory, landing a horizontal distance $D$ from the end of the track (measured from directly below $B$). A note states "Figure not drawn to scale."]
A block of mass $m$ is placed on top of an ideal spring of spring constant $k$. The block is pushed against the spring, compressing the spring a distance $\Delta x$. The block is released from rest, leaves the spring at the position shown in the figure, travels upward, and enters a track with a constant radius of curvature $R$ that has negligible friction. The block enters the track at point $A$, maintains contact with the track, and exits horizontally at point $B$, a distance $3R$ above the point the block was released. The block then falls to the ground and lands a horizontal distance $D$ from the end of the track. Express all algebraic answers in terms of $m$, $k$, $\Delta x$, $R$, and physical constants, as appropriate. The size of the block is much smaller than the radius of curvature of the track.
On the dot below, which represents the block, draw and label the forces (not components) that act on the block while still in contact with the track at point B. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.
[A single black dot is shown, representing the block at point B, with blank space around it for the student to draw force arrows.]
Justify your choice of vectors.
Derive an expression for the speed $v$ of the block at point B.
Derive an expression for the magnitude of the net force $F$ on the block at point B.
Derive an expression for the minimum value of $\Delta x_{\min}$ required in order for the block to maintain contact with the track through point B.
The procedure is repeated several times with the distance $\Delta x > \Delta x_{\min}$.
Calculate the distance $D$ that the block travels.
The graph below shows the best-fit line drawn by the students through their data of $D$ as a function of $\Delta x$.
[Graph: vertical axis $D$, horizontal axis $\Delta x$. A dashed horizontal line marks $D_{\text{MIN}}$ on the $D$-axis, and a dashed vertical line marks $\Delta x_{\min}$ on the $\Delta x$-axis, meeting at the start of the data trend. Region "I" is labelled in the lower-left area (below/left of the dashed lines, near the origin $O$, where there is no plotted line). Region "II" is labelled along a straight line that starts at the point $(\Delta x_{\min}, D_{\text{MIN}})$ and rises linearly up and to the right.]
Explain why there are no data for section I of the graph.
Explain the reason for the shape and minimum value of section II on the graph.