Learn Extracted exam questions AP Physics C: Mechanics 2023 Free Response · Set 2
2023 Free Response · Set 2
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Scientists have created a new type of lightweight foam and are performing experiments to investigate the properties of the foam. The mass of Cart A is 1000 kg and the mass of Cart B is 2000 kg. A piece of foam with negligible mass is attached to the front of Cart A, as shown. Cart A moves with a constant speed toward Cart B, which is initially at rest. At time $t = 0$ s, the foam connected to Cart A makes contact with Cart B. The foam remains in contact with Cart B for 0.5 s, after which the carts separate and both carts move with constant velocities.
[Diagram: Cart A (with a block labeled "Foam" attached to its front) approaching Cart B along a horizontal track, both carts shown as boxes on wheels.]
The graph shows the velocity $v$ of Cart A as a function of time $t$ for the time interval when the foam and Cart B are in contact.
[Graph: $v$ (m/s) vs. $t$ (s); y-axis from 0 to 5 m/s in increments of 1, x-axis from 0 to 0.5 s in increments of 0.1 s; curve starts at $(0, 5)$ and decreases smoothly, concave up, leveling off toward approximately $(0.5, 1)$.]
What feature(s) of the graph could be used to estimate the displacement of Cart A during the collision?
Using the information shown in the graph, determine the speed of Cart B at $t = 0.5$ s.
On the following grid, draw a smooth curve of the velocity of Cart B as a function of time.
[Grid: $v$ (m/s) vs. $t$ (s); y-axis from 0 to 5 m/s in increments of 1, x-axis from 0 to 0.5 s in increments of 0.1 s; the curve for Cart A is already plotted and labeled "Cart A," starting at $(0,5)$ and decreasing to approximately $(0.5,1)$.]
For $0 \le t \le 0.50$ s, the velocity $v$ of Cart A can be described by the function $v(t) = 64t^3 - 48t^2 + 5$.
Calculate the magnitude of the maximum net force acting on Cart A during this interval.
On the following grid, draw a smooth curve of the magnitude of the force acting on Cart A as a function of time. Clearly indicate the value of the maximum force on the vertical axis.
[Grid: $F$ (N) vs. $t$ (s); x-axis from 0 to 0.5 s in increments of 0.1 s; y-axis unlabeled/blank for the student to fill in.]
The foam is removed from the front of Cart A and the experiment is repeated. The carts collide, with both Cart A and Cart B having the same initial and final velocities as in the original collision. The time intervals during which the carts are in contact are different in the collision with the foam and the collision without the foam. In the collision without the foam, Cart A is in contact with Cart B for a shorter duration than in the original collision, when the foam was present.
For the original collision when the foam is present, the magnitude of the average net force exerted on Cart B is $F_1$. For the collision without the foam, the magnitude of the average net force exerted on Cart B is $F_2$.
What is the relationship between the magnitude of the average net force $F_1$ exerted on Cart B for the collision with the foam and the magnitude of the average net force $F_2$ exerted on Cart B for the collision without the foam?
____ $F_1 > F_2$ ____ $F_1 < F_2$ ____ $F_1 = F_2$
Justify your answer.
A student conducts an investigation to determine the relationship between the period of oscillation $T$ of a system consisting of a block and $N$ attached springs. The student starts with a block of mass $m$ attached to a single ideal spring of spring constant $k$, as shown in Figure 1. The student holds the block so that the spring is neither stretched nor compressed at a vertical height 1.00 m above a motion detector. The student releases the block from rest and records the period of oscillation for the system consisting of the single spring and block. An additional identical spring is attached in parallel, as shown in Figure 2, and the procedure is repeated for $N = 2$ springs. This procedure is repeated through $N = 10$ springs.
[Figure 1: A spring of spring constant $k$ (labeled "$N = 1$") hangs from a fixed support, with a block of mass $m$ attached below it, positioned 1.00 m above a motion detector on the ground.]
[Figure 2: Two identical springs, each of spring constant $k$, attached in parallel (labeled "$N = 2$") hang from a fixed support, with a block of mass $m$ attached below them, positioned 1.00 m above a motion detector on the ground.]
Derive an expression for $T$ as a function of $N$. Express your answer in terms of $m$, $k$, $N$, and physical constants as appropriate.
On the following axes, sketch a graph of $T$ as a function of $N$ for $N \ge 1$.
[Grid: axes labeled $T$ (vertical) vs. $N$ (horizontal), both unlabeled with values, blank for the student to sketch on.]
The student plots the data for $T^2$ as a function of $N^{-1}$, as shown.
[Graph: $T^2$ (s$^2$) vs. $N^{-1}$; y-axis from 0 to 12 in increments of 2, x-axis from 0.0 to 1.1 in increments of 0.1; scattered data points roughly along an increasing linear trend, approximately at $(0.1, 1)$, $(0.15, 1.5)$, $(0.2, 2)$, $(0.33, 4)$, $(0.5, 6)$, $(1.0, 12)$.]
Draw the best-fit line for the data.
The mass of the block is measured to be $m = 1.5$ kg. Using the graph, calculate an experimental value for the spring constant $k$ for a single spring.
The student finds that the value given by the manufacturer for the spring constant is larger than the value determined experimentally in part (c)(ii). Determine a single source of experimental error that could result in the observed difference in the value for $k$. Briefly justify your answer.
The student conducts a similar investigation to determine the relationship between the period of oscillation $T$ of a block-spring system and a number $N$ of identical springs horizontally on a table, as shown in Figure 3. Frictional forces between the table and the block are negligible. In each trial, the block is displaced the same horizontal distance from equilibrium and released from rest.
[Figure 3: A block of mass $m$ attached to $N = 2$ springs on a horizontal table, springs shown attached to a wall on the left and the block on the right.]
The student plots $T^2$ as a function of $N^{-1}$ for this new investigation. Would the slope of the best-fit line from this new investigation be greater than, less than, or the same as the slope of the best-fit line in part (c)(i)?
____ greater than ____ less than ____ the same as
Briefly justify your answer.
When $N = 1$, the maximum speed of the block is found to be $v_{max}$. When $N$ increases, will $v_{max}$ increase, decrease, or stay the same?
____ increase ____ decrease ____ stay the same
Justify your answer.
A wind turbine includes a three-blade system that rotates about an axis through the end of each blade, as shown in Figure 1. Each blade has a length $L$ and mass $M$, with a center of mass located at a distance $\dfrac{L}{3}$ from the axis of rotation, as shown in Figure 2.
[Figure 1: "Three-Blade System" — a turbine with three blades extending from a central hub, with the "Axis of Rotation" labeled through the hub and one "Blade" labeled.]
[Figure 2: "Single Blade" — a blade of length $L$ shown horizontally, with the "Axis of Rotation" at one end and the "Center of Mass" marked at a distance $\dfrac{L}{3}$ from the axis of rotation.]
Derive an expression for the rotational inertia of the three-blade system. Express your answer in terms of $M$, $L$, and physical constants, as appropriate. The rotational inertia of each blade about an axis through its center of mass is given by the equation $I_{cm} = \dfrac{1}{18}ML^2$.
While the wind blows, the three-blade system operates at a constant angular speed $\omega_0 = 2.6$ rad/s. The length of one blade is $L = 36$ m. The numerical value of the rotational inertia of the system is $I_{sys} = 6.7 \times 10^6$ kg$\cdot$m$^2$. Calculate the time $T$ it takes the outer edge of a single blade to complete one revolution.
When the wind stops blowing, the angular speed of the system decreases. The angular speed $\omega$ of the system while slowing down is given as a function of time $t$ by the equation $\omega = \omega_0 e^{-\beta_0 t}$, where $\beta_0$ is a constant with appropriate units, as shown on the graph in Figure 3.
[Figure 3: Graph of $\omega$ (rad/s) vs. $t$ (s); y-axis from 0 to 3.0 in increments of 1.0, x-axis from 0 to 35 in increments of 5; curve starts at $(0, 2.6)$ and decays exponentially toward 0 as $t$ increases, with a dashed vertical line at $t=0$ marking the initial value.]
Calculate the amount of energy dissipated from $t = 0$, when the wind stops blowing, until the system comes to rest.
Derive an expression for the net torque exerted on the system as a function of $t$ as the system slows down. Express your answer in terms of $\beta_0$, $\omega_0$, $M$, $L$, $I_{sys}$, and physical constants, as appropriate.
Derive an expression for the angular displacement of the system $\Delta\theta$ as a function of $t$. Express your answer in terms of $\beta_0$, $\omega_0$, $M$, $L$, $I_{sys}$, and physical constants, as appropriate.
The three-blade system is now replaced with a second three-blade system identical to the first, except that the second three-blade system slows down according to the equation $\omega = \omega_0 e^{-\beta t}$, where $\omega_0 = 2.6$ rad/s and $\beta > \beta_0$. The original angular speed function is shown as a dashed line in Figure 4.
[Figure 4: Graph of $\omega$ (rad/s) vs. $t$ (s); y-axis from 0 to 3.0 in increments of 1.0, x-axis from 0 to 35 in increments of 5; a dashed curve shows the original $\omega_0 e^{-\beta_0 t}$ decay from $(0,2.6)$ toward 0.]
On the graph in Figure 4, sketch the angular speed of the second three-blade system as a function of time $t$.