A small punching bag for boxing workouts is approximately the size and weight of a person's head and is suspended from a very short rope or chain. Estimate the natural frequency of oscillations of such a punching bag.

Two damped, driven spring-mass oscillating systems have identical driving forces as well as identical spring and damping constants. However, the mass of system $A$ is four times the mass of system B. Assume both systems are very weakly damped. How do their resonant frequencies compare? $(a)$ $\omega_{\text{A}}=\omega_{\text{B}}$ $(b)$ $\omega_{\text{A}}=2 \omega_{\text{B}}$ , $(c)$ $\omega_{\text{A}}=\frac{1}{2} \omega_{\text{B}}$,$(d)$ $\omega_{\text{A}}=\frac{1}{4} \omega_{\text{B}}$,$(e)$ Their resonant frequencies cannot be compared, given the information provided.

Two damped spring-mass oscillating systems have identical spring constants and decay times. However, system $A^{\prime}$ s mass $m_{\text{A}}$ is twice system B's mass $m_{\text{B}}$. How do their damping constants, $b$, compare? $(a)$ $b_{\text{A}}=4 b_{\text{B}}$,$(b)$ $b_{\text{A}}=2 b_{\text{B}}$,$(c)$ $b_{\text{A}}=b_{\text{B}}$,$(d)$ $b_{\text{A}}=\frac{1}{2} b_{\text{B}}$, $(e)$ Their decay times cannot be compared, given the information provided.

Two damped spring-mass oscillating systems have identical spring and damping constants. However, system A's mass $m_{\text{A}}$ is four times system $B^{\prime}$ s mass $m_{\text{B}}$. How do their decay times compare? $(a)$ $\tau_{\text{A}}=4 \tau_{B^{\prime}}$ $(b)$ $\tau_{\text{A}}=2 \tau_{B^{\prime}}$ $(c)$ $\tau_{\text{A}}=\tau_{B^{\prime}}$ $(d)$ Their decay times cannot be compared, given the information provided.

If the amplitude of a simple harmonic oscillator is tripled, by what factor is the energy changed?

"True or false: $(c)$ If the net force on a particle undergoing one-dimensional motion is proportional to, and oppositely directed from, the displacement from equilibrium, the motion is simple harmonic."