AP Physics C Mechanics Quiz: Representing And Analyzing Shm

A frictionless mass–spring oscillator has $m=0.10\ \text{kg}$, $k=40\ \text{N/m}$, and equilibrium at $x=0$. It is released from rest at $x=+0.050\ \text{m}$ at $t=0$, so $A=0.050\ \text{m}$ and $\phi=0$. The motion is $x(t)=A\cos(\omega t)$ with $\omega=\sqrt{k/m}$ and period $T=2\pi/\omega$. The object reaches equilibrium after one quarter period. All quantities use SI units and amplitude remains constant. How long after release does it first reach $x=0$?

A 0.30 kg mass on a spring with $k=120\ \text{N/m}$ oscillates without friction about equilibrium $x=0$. At $t=0$, the mass passes through equilibrium moving in the positive direction with speed $0.60\ \text{m/s}$. Using $x(t)=A\cos(\omega t+\phi)$ and $v(t)=-A\omega\sin(\omega t+\phi)$ with $\omega=\sqrt{k/m}$, the condition $x(0)=0$ implies $\cos\phi=0$. The sign of $v(0)$ sets the phase quadrant. All values are SI units and amplitude is constant. What is the phase constant $\phi$?

A 0.40 kg block attached to a spring with $k=160\ \text{N/m}$ oscillates on a frictionless surface about $x=0$. At $t=0$ it is at $x=+0.050\ \text{m}$ moving in the negative direction with speed $0.40\ \text{m/s}$. For SHM, $x(t)=A\cos(\omega t+\phi)$ with $\omega=\sqrt{k/m}$ and $v(t)=-A\omega\sin(\omega t+\phi)$. The amplitude satisfies $A^2=x_0^2+(v_0/\omega)^2$ using SI units. The phase $\phi$ is determined by $x_0$ and $v_0$, but $A$ is independent of sign conventions. Mechanical energy is conserved. What is the amplitude $A$?

A simple pendulum has length $L=0.80\ \text{m}$ and bob mass $m=0.15\ \text{kg}$, oscillating with small angle so SHM applies. Take $g=9.8\ \text{m/s}^2$ and equilibrium at $\theta=0$. The angular frequency is $\omega=\sqrt{g/L}$ and the motion can be written $\theta(t)=\theta_{\max}\cos(\omega t+\phi)$ with $\theta$ in radians. The period is $T=2\pi/\omega$ and the frequency is $f=1/T$. The phase $\phi$ is set by initial conditions, but does not affect $T$. Air resistance is negligible, so amplitude stays constant. What is the period $T$ of the pendulum?

A 0.60 kg cart is attached to a spring with $k=240\ \text{N/m}$ and oscillates on a frictionless track about $x=0$. It is pulled to $x=+0.070\ \text{m}$ and released from rest at $t=0$, so $A=0.070\ \text{m}$ and $\phi=0$. The motion is $x(t)=A\cos(\omega t+\phi)$ with $\omega=\sqrt{k/m}$. The acceleration is $a(t)=-\omega^2 x(t)$ and $a_{\max}=\omega^2 A$. All quantities are in SI units and amplitude remains constant. Find the maximum acceleration magnitude $a_{\max}$.

A 0.20 kg mass is attached to a horizontal spring with $k=50\ \text{N/m}$ on a frictionless surface, with equilibrium at $x=0$. The mass is displaced to $x=+0.060\ \text{m}$ and released from rest at $t=0$, so $A=0.060\ \text{m}$ and $\phi=0$. The SHM model is $x(t)=A\cos(\omega t+\phi)$, where $\omega=\sqrt{k/m}$ in rad/s. The speed is $v(t)=-A\omega\sin(\omega t+\phi)$ and the maximum speed occurs at $x=0$. The restoring force is $F_s=-kx$ and mechanical energy is conserved. What is the maximum speed at equilibrium?

A small-angle pendulum has length $L=1.2\ \text{m}$ and bob mass $m=0.20\ \text{kg}$, swinging about $\theta=0$ with constant amplitude $\theta_{\max}$. Take $g=9.8\ \text{m/s}^2$ and model the motion as $\theta(t)=\theta_{\max}\cos(\omega t+\phi)$ with $\omega=\sqrt{g/L}$. The period is $T=2\pi/\omega$ and the frequency is $f=1/T$. The phase $\phi$ depends on initial conditions but does not change $f$. Assume no air resistance and SI units. Determine the frequency $f$.

A mass–spring oscillator has $m=0.50\ \text{kg}$, $k=80\ \text{N/m}$, and equilibrium at $x=0$ on a frictionless surface. It is displaced to $A=0.10\ \text{m}$ and released from rest, so $x(t)=A\cos(\omega t)$ with $\omega=\sqrt{k/m}$. The total energy is constant and given by $E=\tfrac12 kA^2$ in joules. At equilibrium the kinetic energy equals $E$ and the spring potential energy is zero. Use SI units throughout and ignore any damping. What is the total mechanical energy $E$?

A horizontal mass–spring system has $m=0.25\ \text{kg}$, $k=100\ \text{N/m}$, and equilibrium at $x=0$. The mass is released from rest at $x=+0.040\ \text{m}$ at $t=0$, so $A=0.040\ \text{m}$ and $\phi=0$. The SHM equation is $x(t)=A\cos(\omega t+\phi)$ with $\omega=\sqrt{k/m}$. The period is $T=2\pi/\omega$ and the frequency is $f=1/T$. The phase affects where the motion starts but not $f$. Assume no friction and constant amplitude. Determine the frequency $f$ of oscillation.

A 0.50 kg block on a frictionless track is attached to a spring with $k=200\ \text{N/m}$ and equilibrium at $x=0$. It is pulled to $x=+0.080\ \text{m}$ and released from rest at $t=0$, so $A=0.080\ \text{m}$ and $\phi=0$. The angular frequency is $\omega=\sqrt{k/m}$ and the motion is $x(t)=A\cos(\omega t+\phi)$ with $x$ in meters and $t$ in seconds. The velocity is $v(t)=-A\omega\sin(\omega t+\phi)$ and the acceleration is $a(t)=-\omega^2 x(t)$. The restoring force is $F_s=-kx$. Assume SHM holds for all times. What is the period $T$ of the oscillation?

A mass $m$ is attached to two identical springs, each with spring constant $k$, arranged in parallel (both attached to the mass and to fixed supports). The system oscillates with simple harmonic motion. If one spring suddenly breaks when the mass is at maximum displacement $+A$, what will be the new amplitude of oscillation?

A mass attached to a spring oscillates with simple harmonic motion. At time $t = 0$, the mass is at position $x = +A/2$ and moving in the positive direction with speed $v = \frac{\sqrt{3}}{2}\omega A$, where $A$ is the amplitude and $\omega$ is the angular frequency. What is the phase constant $\phi$ in the equation $x(t) = A\cos(\omega t + \phi)$?

Two pendulums of equal length $L$ are released simultaneously from small angular displacements. Pendulum A has mass $m_A$ concentrated at the bob, while Pendulum B has the same total mass $m_A$ distributed as a uniform rod of length $L$ (no additional bob). What is the ratio of their periods $T_A/T_B$?

A horizontal spring-mass system has a block of mass $m$ attached to a spring with constant $k$. The block is displaced from equilibrium and released. If the coefficient of kinetic friction between the block and surface is $\mu_k$, which expression best represents the motion during the first quarter period after release?

A particle undergoes simple harmonic motion with position $x(t) = 0.25\cos(4t + \pi/4)$ meters. At what time $t > 0$ does the particle first reach maximum kinetic energy?