Question 1

Which of the following best describes the equation of motion for a simple harmonic oscillator with displacement $$x(t)$$?

Accepted Answer

$$m \frac{d^2$x}{dt^2$} + kx = 0$$

Question 2

A mass $$m$$ attached to a spring with spring constant $$k$$ oscillates with period $$T. $$Which formula gives the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{m}{k}}$$

Question 3

For a simple pendulum of length $$l$$ under gravity $$g$$, what is the angular frequency $$\omega$$ for small oscillations?

Accepted Answer

$$\omega = \sqrt{\frac{g}{l}}$$

Question 4

A physical pendulum consists of a rigid body of mass $$M$$ and moment of inertia $$I$$ about the pivot. The distance from the pivot to the center of mass is $$d. $$What is the period $$T$$ for small oscillations?

Accepted Answer

$$T = 2\pi \sqrt{\frac{I}{Mgd}}$$

Question 5

A damped oscillator is described by $$m \frac{d^2$x}{dt^2$} + b \frac{dx}{dt} + kx = 0. $$What happens to the amplitude over time if the system is underdamped?

Accepted Answer

The amplitude decreases exponentially but oscillations persist.

Question 6

Which of the following is the general solution for the displacement $$x(t) of a $$simple harmonic oscillator?

Accepted Answer

$$x(t) = A \cos(\omega t + \phi)$$

Question 7

A mass-spring system oscillates with amplitude $$A$$ and angular frequency $$\omega. $$What is the maximum speed of the mass?

Accepted Answer

$$v_{max} = A \omega$$

Question 8

For a simple pendulum, which assumption is necessary for the period formula $$T = 2\pi \sqrt{\frac{l}{g}} to be $$valid?

Accepted Answer

The oscillations are small (small angle approximation).

Question 9

A uniform rod of length $$L$$ and mass $$M is $$pivoted at one end. What is the period of small oscillations?

Accepted Answer

$$T = 2\pi \sqrt{\frac{2L}{3g}}$$

Question 10

Which of the following best describes the restoring force in a simple harmonic oscillator?

Accepted Answer

$$F = -kx$$

Question 11

A damped oscillator has a damping coefficient $$b$$ and mass $$m. $$What is the condition for critical damping?

Accepted Answer

$$b^2 = 4mk$

Question 12

A mechanical wave travels along a string with speed $$v. If $$the tension in the string is $$T$$ and the mass per unit length is $$\mu$$, what is the wave speed?

Accepted Answer

$$v = \sqrt{\frac{T}{\mu}}$$

Question 13

Which of the following is true for the energy in a simple harmonic oscillator?

Accepted Answer

Total energy is constant and equals $$\frac{1}{2}kA^2$

Question 14

A mass $$m$$ attached to a spring is displaced and released from rest at $$x = A. $$What is the velocity when it passes through equilibrium?

Accepted Answer

$$v = \omega A$$

Question 15

Which of the following best describes resonance in a driven oscillator?

Accepted Answer

The amplitude becomes maximum when the driving frequency matches the natural frequency.

Question 16

A wave on a string is described by $$y(x,t) = A \sin(kx - \omega t). $$What is the phase velocity of the wave?

Accepted Answer

$$v = \frac{\omega}{k}$$

Question 17

If the amplitude of a damped oscillator drops from $$A_0 to$$ $$A_0/2$ in time t$, what is the logarithmic decrement $$\delta$$?

Accepted Answer

$$\delta = \ln 2$$

Question 18

Which of the following is the correct expression for the kinetic energy of a simple harmonic oscillator at displacement $$x$$?

Accepted Answer

$$K = \frac{1}{2}k(A^2$ - x^2$)$$

Question 19

A sound wave travels in air at STP. What is the approximate speed of sound?

Accepted Answer

$$343\ \mathrm{m/s}$$

Question 20

Which of the following statements about mechanical waves is correct?

Accepted Answer

Mechanical waves require a medium to propagate.

Oscillations and Mechanical Waves: Study Notes

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