AP Physics C Mechanics - 7.5 Simple and Physical Pendulums- Exam Style questions- FRQs

Simple and Physical Pendulums AP  Physics C Mechanics FRQ

Unit 7: Oscillations

Weightage : 10-15%

AP Physics C Mechanics Exam Style Questions – All Topics

Question

Block A and Block B of masses $m$ and $3m$, respectively, are arranged in a setup consisting of an ideal spring with spring constant $k$ and a horizontal surface. Friction between the surface and the blocks is negligible except in a region of length $D$, where the coefficient of kinetic friction between Block A and the surface is $\mu$. Block B is attached to a string of length $\ell$ and negligible mass, as shown in Figure 1. Block A is held against the spring, compressing the spring a distance $x_{c}$.
At time $t=0$ Block A is located at position $x=x_{0}$ and is released from rest. After the block is released, the following occurs.
• At time $t=t_{1}$, Block A is at $x=x_{1}$ after traveling a distance $x_{c}$. Block A moves with speed $v$, and the spring is at its equilibrium position.
• At time $t=t_{2}$ the left side of Block A is at $x=x_{2}$ after passing through a distance $D$ across the region with nonnegligible friction.
• At time $t=t_{3}$, Block A is at $x=x_{3}$ and Block A collides with and sticks to Block B.
(a) For parts (a)(i) and (a)(ii), express your answer in terms of $m$, $k$, $D$, $\mu$, $x_{c}$, and physical constants, as appropriate.
i. Derive an expression for the speed $v$ of Block A at time $t_{1}$.
ii. Derive an expression for the speed $v_{A,B}$ of the two-block system immediately after the collision at time $t_{3}$.
(b)
i. Sketch a graph of the kinetic energy $K$ of Block A as a function of time $t$ from time $t=0$ to time $t_{3}$.

ii. Use principles of work and energy to justify the graph drawn in part (b)(i) for the time interval $t=0$ to $t=t_{1}$. Explicitly reference features of the shape of the graph you drew in part (b)(i).
After the collision, the two-block system instantaneously comes to rest at time $t_{4}$, which occurs when the string makes a small angle $\theta_{max}$ with the vertical. For times $t>t_{4}$, the system oscillates with frequency $f_{1}$. The support holding the string is raised, and the procedure is then repeated using a new string of length $2\ell$.
(c) Indicate how the new frequency of oscillation $f_{2\ell}$ of the system on the new string of length $2\ell$ will compare to the frequency of oscillation $f_{1}$ from the original procedure.
_____ $f_{2\ell} > f_{1}$ _____ $f_{2\ell} < f_{1}$ _____ $f_{2\ell} = f_{1}$
Briefly justify your answer.

Most-appropriate topic codes (AP Physics C: Mechanics):

• Topic 3.2 — Work (Parts a, b)
• Topic 3.4 — Conservation of Energy (Parts a, b)
• Topic 4.4 — Elastic and Inelastic Collisions (Part a)
• Topic 7.5 — Simple and Physical Pendulums (Part c)
▶️ Answer/Explanation

(a)(i)
Using conservation of energy, the stored spring potential energy converts into kinetic energy since there is no friction in this first section.

$U_{s} = K$
$\frac{1}{2}k x_{c}^{2} = \frac{1}{2}m v^{2}$
$v = x_{c}\sqrt{\frac{k}{m}}$

(a)(ii)
Block A loses kinetic energy due to the negative work done by friction over distance $D$.

$K_{after\_friction} = K_{initial} – W_{friction}$
$\frac{1}{2}m v_{2}^{2} = \frac{1}{2}k x_{c}^{2} – \mu m g D$
$v_{2} = \sqrt{\frac{k x_{c}^{2}}{m}{ – 2 \mu g D}}$

In the perfectly inelastic collision, Block A ($m$) sticks to Block B ($3m$). Conservation of momentum gives the final speed.

$m v_{2} = (m + 3m)v_{A,B}$
$v_{A,B} = \frac{v_{2}}{4}$
$v_{A,B} = \frac{1}{4}\sqrt{\frac{k x_{c}^{2}}{m}{ – 2 \mu g D}}$

(b)(i)


Between $t=0$ and $t_{1}$, the $K$ curve starts at zero with a flat slope, smoothly curves upward, and flattens out to a horizontal peak at $t_{1}$ (creating a wave-like $S$-curve or $\sin^2$ shape). Between $t_{1}$ and $t_{2}$, $K$ stays constant (a straight horizontal line). During the friction zone starting at $t_{2}$, the $K$ curve drops down shaped like an upward-opening parabola to a lower constant value until $t_{3}$.

(b)(ii)
By the work-energy theorem, the change in kinetic energy equals the work done by the spring. The power (the rate of work done, and thus the slope of the $K$ vs $t$ graph) is $P = Fv$. At $t=0$, $v=0$, so the initial slope is exactly zero. As the block speeds up, the power increases, but as it nears equilibrium, the spring force $F$ drops to zero. Because $F=0$ at equilibrium, the slope of the graph gradually flattens back out to zero at $t_{1}$.

(c)
$f_{2\ell} < f_{1}$

Once they stick together and swing, the two blocks act as a simple pendulum. A simple pendulum’s frequency is given by $f = \frac{1}{2\pi}\sqrt{\frac{g}{\ell}}$. Since the frequency is inversely proportional to the square root of the string’s length, increasing the length to $2\ell$ decreases the natural oscillation frequency.

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