Question 1

A mass $$m is $$attached to a spring with spring constant $$k$$ and allowed to oscillate horizontally on a frictionless surface. What is the angular frequency $$\$$omega_0$$ of the system?

Accepted Answer

$$\$$omega_0$$ = \sqrt{\frac{k}{m}}$$

Question 2

A 0.5 kg mass is attached to a spring with $$k = 200\ \mathrm{N/m}$$ and a damping coefficient $$b = 1\ \mathrm{kg/s}. Is $$the system underdamped, overdamped, or critically damped?

Accepted Answer

Underdamped

Question 3

For a damped harmonic oscillator, which of the following correctly expresses the quality factor $$Q$$?

Accepted Answer

$$Q = \frac{m \$$omega_0$$}{b}$$

Question 4

A mass-spring system is subject to a driving force $$F(t) = F_0$ \cos(\omega t). At $$resonance, what is the amplitude of oscillation?

Accepted Answer

$$A_{res} = \frac{F_0$}{b \$$omega_0$$}$$

Question 5

Which of the following statements about the effect of damping on the frequency of a harmonic oscillator is correct?

Accepted Answer

Damping decreases the oscillation frequency compared to the undamped case.

Question 6

Given a mass $$m$$ attached to a spring with constant $$k$$ and a damper with coefficient $$b$$, what is the condition for critical damping?

Accepted Answer

$$b = 2\sqrt{km}$$

Question 7

A block of mass $$m is $$attached to a spring and oscillates with period $$T. If $$the mass is quadrupled, what happens to the period?

Accepted Answer

The period doubles: $$T' = 2T$$

Question 8

For a damped oscillator, the displacement as a function of time is given by $$x(t) = A e$$^{-\gamma t}$$ \cos(\omega' t + \phi). $$What does $$\gamma$$ represent?

Accepted Answer

The damping coefficient per unit mass: $$\gamma = \frac{b}{2m}$$

Question 9

A mass-spring-damper system is driven at a frequency much higher than its natural frequency. What happens to the amplitude of steady-state oscillations?

Accepted Answer

The amplitude approaches zero as frequency increases.

Question 10

Which of the following best describes the solution to the equation $$m \ddot{x} + b \dot{x} + kx = 0$$ for the overdamped case?

Accepted Answer

A sum of two decaying exponentials with different rates.

Question 11

A block of mass $$m is $$attached to a spring with constant $$k$$ and is displaced from equilibrium and released. Which of the following is the correct equation for the period $$T of $$oscillation?

Accepted Answer

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

Question 12

In a forced, damped oscillator, at what driving frequency does resonance occur?

Accepted Answer

At $$\omega = \sqrt{\$$omega_0$$^2 - 2\$$gamma^2$$}$$

Question 13

A mass-spring system is critically damped. If displaced and released, what is the behavior of the system?

Accepted Answer

It returns to equilibrium as quickly as possible without oscillating.

Question 14

Which of the following is true for the amplitude $$A of a $$driven, damped oscillator at resonance?

Accepted Answer

$$A is $$inversely proportional to the damping coefficient $$b.$$

Question 15

A block of mass $$m is $$attached to a spring and oscillates with period $$T. If $$the spring constant $$k is $$increased by a factor of 4, what happens to the period?

Accepted Answer

The period is halved: $$T' = \frac{T}{2}$$

Question 16

For a simple pendulum of length $$l$$ and small amplitude, what is the period $$T$$?

Accepted Answer

$$T = 2\pi \sqrt{\frac{l}{g}}$$

Question 17

Which of the following statements about the steady-state amplitude of a driven, damped oscillator is correct?

Accepted Answer

It is maximum when the driving frequency matches the resonance frequency.

Question 18

A mass-spring system is released from rest at $$x_0$$ with no initial velocity. What is the general solution for its displacement in the absence of damping?

Accepted Answer

$$x(t) = x_0$ \cos(\$$omega_0$$ t)$$

Question 19

Which of the following is the correct expression for the angular frequency of a damped oscillator?

Accepted Answer

$$\omega' = \sqrt{\$$omega_0$$^2 - \$$gamma^2$$}$$

Study Guide: Periodic Motion and Damped Oscillations

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