Learn Extracted exam questions AP Physics C: Mechanics 2014 Free Response
2014 Free Response
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[Figure: A horizontal track marked in centimeters from 0 to 120 cm. A cart with a spring attached to the left end of the track sits near the 0 cm mark, with two wheels shown. Five vertical photogates labelled 1, 2, 3, 4, 5 are mounted on the track at positions 20 cm, 40 cm, 60 cm, 80 cm, and 100 cm respectively, labelled "Photogates" above.]
In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal, level track. The spring is attached to the left end of the track, as shown in the figure above, and produces a nonlinear restoring force of magnitude $F_s = As^2 + Bs$, where $s$ is the distance the spring is compressed, in meters. A measuring tape, marked in centimeters, is attached to the side of the track. The student places five photogates on the track at the locations shown.
Derive an expression for the potential energy $U$ as a function of the compression $s$. Express your answer in terms of $A$, $B$, $s$, and fundamental constants, as appropriate.
In a preliminary experiment, the student pushes the cart of mass 0.30 kg into the spring, compressing the spring 0.040 m. For this spring, $A = 200\ \text{N/m}^2$ and $B = 150\ \text{N/m}$. The cart is released from rest. Assume friction and air resistance are negligible only during the short time interval when the spring is accelerating the cart.
Calculate the following: The speed of the cart immediately after it loses contact with the spring
Calculate the following: The impulse given to the cart by the spring
In a second experiment, the student collects data using the photogates. Each photogate measures the speed of the cart as it passes through the gate. The student calculates a spring compression that should give the cart a speed of $0.320\ \text{m/s}$ after the cart loses contact with the spring. The student runs the experiment by pushing the cart into the spring, compressing the spring the calculated distance, and releasing the cart. The speeds are measured with a precision of $\pm 0.002\ \text{m/s}$. The positions are measured with a precision of $\pm 0.005\ \text{m}$.
| Photogate | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Cart speed (m/s) | 0.412 | 0.407 | 0.399 | 0.374 | 0.338 |
| Photogate position (m) | 0.20 | 0.40 | 0.60 | 0.80 | 1.00 |
On the axes below, plot the data points for the speed $v$ of the cart as a function of position $x$. Clearly scale and label all axes, as appropriate.
[Blank grid of axes for plotting speed $v$ vs. position $x$, ruled in a 4x4 arrangement of major squares each subdivided into a 5x5 fine grid, no pre-printed scale numbers or labels.]
Compare the speed of the cart measured by photogate 1 to the predicted value of the speed of the cart just after it loses contact with the spring. List a physical source of error that could account for the difference.
From the measured speed values of the cart as it rolls down the track, give a physical explanation for any trend you observe.
[Figure, Side View: A small block of mass $m$ sits at the top of a ramp of height $h$ above a horizontal tabletop; a dashed horizontal line runs from the base of the ramp to point $P$ at the bottom right of the ramp, where a circular vertical wall (viewed edge-on, shown as a band/collar shape) is mounted on the tabletop.]
[Figure, Top View: A "Rough Wall" is drawn as a large circle (with a curved arrow indicating the wall curls around) of radius $R$ (dashed line from the circle's center area to the wall, labelled $R$), connected to a "Smooth Ramp" (rectangular block) that leads into the circular wall at point $P$.]
Mech. 2.
A small block of mass $m$ starts from rest at the top of a frictionless ramp, which is at a height $h$ above a horizontal tabletop, as shown in the side view above. The block slides down the smooth ramp and reaches point $P$ with a speed $v_0$. After the block reaches point $P$ at the bottom of the ramp, it slides on the tabletop guided by a circular vertical wall with radius $R$, as shown in the top view. The tabletop has negligible friction, and the coefficient of kinetic friction between the block and the circular wall is $\mu$.
Derive an expression for the height of the ramp $h$. Express your answer in terms of $v_0$, $m$, and fundamental constants, as appropriate.
A short time after passing point $P$, the block is in contact with the wall and moves with a speed of $v$.
Is the vertical component of the net force on the block upward, downward, or zero?
____ Upward ____ Downward ____ Zero
Justify your answer.
[Figure: Top view showing a circular wall (partial arc drawn) with a straight horizontal track leading into it from the left; a small block (filled square) is shown at the rightmost point of the circle where the track meets the circle, with a dashed vertical line through that point.]
On the figure below, draw an arrow starting on the block to indicate the direction of the horizontal component of the net force on the moving block when it is at the position shown.
Justify your answer.
Express your answers to the following in terms of $v_0$, $v$, $m$, $R$, $\mu$, and fundamental constants, as appropriate.
Determine an expression for the magnitude of the normal force $N$ exerted on the block by the circular wall as a function of $v$.
Derive an expression for the magnitude of the tangential acceleration of the block at the instant the block has attained a speed of $v$.
Derive an expression for $v(t)$, the speed of the block as a function of time $t$ after passing point $P$ on the track.
[Figure, Top View: A large filled circle of radius $R$ (labelled with an arrow from center to edge) representing a disk, with a small stone of mass $m/20$ shown just outside the edge of the disk moving away with velocity $v_0$ (arrow labelled $v_0$, $m/20$).]
[Figure, Side View: A person holding a stone stands atop a raised platform of height $h$ above the ground, throwing the stone horizontally with initial speed $v_0$ (arrow labelled $v_0$); $h$ is marked as the vertical distance from the platform surface to the ground.]
Mech. 3.
A large circular disk of mass $m$ and radius $R$ is initially stationary on a horizontal icy surface. A person of mass $m/2$ stands on the edge of the disk. Without slipping on the disk, the person throws a large stone of mass $m/20$ horizontally at initial speed $v_0$ from a height $h$ above the ice in a radial direction, as shown in the figures above. The coefficient of friction between the disk and the ice is $\mu$. All velocities are measured relative to the ground. The time it takes to throw the stone is negligible. Express all algebraic answers in terms of $m$, $R$, $v_0$, $h$, $\mu$, and fundamental constants, as appropriate.
Derive an expression for the length of time it will take the stone to strike the ice.
Assuming that the disk is free to slide on the ice, derive an expression for the speed of the disk and person immediately after the stone is thrown.
Derive an expression for the time it will take the disk to stop sliding.
[Figure, Top View: A large filled circle (disk) with a small open circle marking its center, and a small stone (filled circle) at the edge of the disk moving tangentially with velocity $v_0$ (arrow labelled $v_0$ pointing tangent to the circle at the edge point).]
The person now stands on a similar disk of mass $m$ and radius $R$ that has a fixed pole through its center so that it can only rotate on the ice. The person throws the same stone horizontally in a tangential direction at initial speed $v_0$, as shown in the figure above. The rotational inertia of the disk is $mR^2/2$.
Derive an expression for the angular speed $\omega$ of the disk immediately after the stone is thrown.
The person now stands on the disk at rest $R/2$ from the center of the disk. The person now throws the stone horizontally with a speed $v_0$ in the same direction as in part (d). Is the angular speed of the disk immediately after throwing the stone from this new position greater than, less than, or equal to the angular speed found in part (d)?
____ Greater than ____ Less than ____ Equal to
Justify your answer.