Learn Extracted exam questions AP Physics C: Mechanics 2018 Free Response
2018 Free Response
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A student wants to determine the value of the acceleration due to gravity $g$ for a specific location and sets up the following experiment. A solid sphere is held vertically a distance $h$ above a pad by an electromagnet, as shown in the figure above. The experimental equipment is designed to release the sphere when the electromagnet is turned off. A timer also starts when the electromagnet is turned off, and the timer stops when the sphere lands on the pad.
[Diagram: A vertical stand on a table. An electromagnet is mounted at the top of the stand, holding a sphere directly beneath it. The distance from the sphere down to a pad on the table is labeled $h$. Labels: "Electromagnet", "Sphere", "$h$" (vertical arrow), "Pad", "Table".]
While taking the first data point, the student notices that the electromagnet actually releases the sphere after the timer begins. Would the value of $g$ calculated from this one measurement be greater than, less than, or equal to the actual value of $g$ at the student's location?
____ Greater than ____ Less than ____ Equal to
Justify your answer.
The electromagnet is replaced so that the timer begins when the sphere is released. The student varies the distance $h$. The student measures and records the time $\Delta t$ of the fall for each particular height, resulting in the following data table.
| $h$ (m) | 0.10 | 0.20 | 0.60 | 0.80 | 1.00 |
|---|---|---|---|---|---|
| $\Delta t$ (s) | 0.105 | 0.213 | 0.342 | 0.401 | 0.451 |
Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for $g$.
Vertical axis: ____________
Horizontal axis: ____________
Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given in the table. Label each row you use and include units.
Plot the data points for the quantities indicated in part (b) on the graph below. Clearly scale and label all axes, including units if appropriate. Draw a straight line that best represents the data.
[Graph: A blank grid (approximately 6 columns × 16 rows of small squares) provided for the student to plot data and draw a best-fit straight line. No axes, scale, or data are pre-printed.]
Using the straight line, calculate an experimental value for $g$.
Another student fits the data in the table to a quadratic equation. The student's equation for the distance fallen $y$ as a function of time $t$ is $y = At^2 + Bt + C$, where $A = 5.75 \text{ m/s}^2$, $B = -0.524 \text{ m/s}$, and $C = +0.080 \text{ m}$. Vertically down is the positive direction.
Using the student's equation above, derive an expression for the velocity of the sphere as a function of time.
Using the student's equation above, calculate the new experimental value for $g$.
Using $9.81 \text{ m/s}^2$ as the accepted value for $g$ at this location, calculate the percent error for the value found in part (e)ii.
Assuming the sphere is at a height of 1.40 m at $t = 0$, calculate the velocity of the sphere just before it strikes the pad.
Two carts are on a horizontal, level track of negligible friction. Cart 1 has a sensor that measures the force exerted on it during a collision with cart 2, which has a spring attached. Cart 1 is moving with a speed of $v_0 = 3.00 \text{ m/s}$ toward cart 2, which is at rest, as shown in the figure above. The total mass of cart 1 and the force sensor is 0.500 kg, the mass of cart 2 is 1.05 kg, and the spring has negligible mass. The spring has a spring constant of $k = 130 \text{ N/m}$. The data for the force the spring exerts on cart 1 are shown in the graph below. A student models the data as the quadratic fit $F = \left(3200 \text{ N/s}^2\right)t^2 - (500 \text{ N/s})t$.
[Diagram: Cart 1 (mass $m = 0.500$ kg) with a Force Sensor attached at its right side, moving right with $v_0 = 3.00 \text{ m/s}$ toward a spring ($k = 130$ N/m) connected to Cart 2 (mass $m = 1.05$ kg, initially at rest, $v = 0$), all on a horizontal track.]
[Graph: Force $F$ (N) on the vertical axis (marked at 5, 0, $-5$, $-10$, $-15$, $-20$, $-25$) versus time $t$ (s) on the horizontal axis (marked 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18). Data points start near $F = 0$ at small $t$, dip down to a minimum of about $-20$ N around $t = 0.08$–$0.10$ s, then rise back up toward $F = 0$ by about $t = 0.16$–$0.18$ s, tracing a roughly parabolic (U-shaped, inverted) curve consistent with the quadratic fit given.]
Using integral calculus, calculate the total impulse delivered to cart 1 during the collision.
Calculate the speed of cart 1 after the collision.
In which direction does cart 1 move after the collision?
____ Left ____ Right
____ The direction is undefined, because the speed of cart 1 is zero after the collision.
Calculate the speed of cart 2 after the collision.
Show that the collision between the two carts is elastic.
Calculate the speed of the center of mass of the two-cart-spring system.
Calculate the maximum elastic potential energy stored in the spring.
A triangular rod, shown above, has length $L$, mass $M$, and a nonuniform linear mass density given by the equation $\lambda = \dfrac{2M}{L^2}x$, where $x$ is the distance from one end of the rod.
Using integral calculus, show that the rotational inertia of the rod about its left end is $ML^2/2$.
[Figure 1: A plain circle (thin hoop), unlabeled interior.]
[Figure 2: A circle (thin hoop) with three identical rods fastened inside it, each rod running from the center of the hoop out to the rim, spaced so the three rods form a symmetric tripod/Y-like pattern (three line segments from the center to three points on the circle).]
The thin hoop shown above in Figure 1 has a mass $M$, radius $L$, and a rotational inertia around its center of $ML^2$. Three rods identical to the rod from part (a) are now fastened to the thin hoop, as shown in Figure 2 above.
Derive an expression for the rotational inertia $I_{tot}$ of the hoop-rods system about the center of the hoop. Express your answer in terms of $M$, $L$, and physical constants, as appropriate.
The hoop-rods system is initially at rest and held in place but is free to rotate around its center. A constant force $F$ is exerted tangent to the hoop for a time $\Delta t$.
Derive an expression for the final angular speed $\omega$ of the hoop-rods system. Express your answer in terms of $M$, $L$, $F$, $\Delta t$, and physical constants, as appropriate.
[Diagram: The hoop-rods system (circle with three internal rods forming a Y-pattern) sitting at the base of an inclined ramp, with a curved arrow above/left of the hoop indicating a force $F$ applied tangent to the hoop, pushing it up the ramp. The ramp rises from left to right, and the hoop sits at the bottom, touching the horizontal surface to the left and the incline surface.]
The hoop-rods system is rolling without slipping along a level horizontal surface with the angular speed $\omega$ found in part (c). At time $t = 0$, the system begins rolling without slipping up a ramp, as shown in the figure above.
On the figure of the hoop-rods system below, draw and label the forces (not components) that act on the system. Each force must be represented by a distinct arrow starting at, and pointing away from, the point at which the force is exerted on the system.
[Diagram: The hoop-rods system (circle with three internal rods in a Y-pattern) resting on a dashed line representing an inclined surface, for the student to draw force vectors on.]
Justify your choice for the direction of each of the forces drawn in part (d)i.
Derive an expression for the change in height of the center of the hoop from the moment it reaches the bottom of the ramp until the moment it reaches its maximum height. Express your answer in terms of $M$, $L$, $I_{tot}$, $\omega$, and physical constants, as appropriate.