Question 1

A mass $$m is $$attached to a spring with spring constant $$k on a $$frictionless surface. If the mass is displaced from equilibrium and released, which of the following equations correctly describes its position as a function of time for simple harmonic motion?

Accepted Answer

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

Question 2

The period $$T of a $$simple pendulum of length $$L ($$assuming small oscillations) is given by which of the following expressions?

Accepted Answer

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

Question 3

A car's suspension system can be modeled as a damped harmonic oscillator. If the damping constant $$b is $$increased, what happens to the amplitude of oscillations after the car hits a bump?

Accepted Answer

The amplitude decreases more rapidly over time.

Question 4

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

Accepted Answer

The amplitude of oscillation is maximized when the driving frequency matches the natural frequency.

Question 5

A block of mass $$m is $$attached to a spring with spring constant $$k$$ and oscillates with amplitude $$A. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2} k A^2$

Question 6

A simple pendulum is released from rest at an angle $$\$$theta_0$$ (small angle). What is the maximum speed of the pendulum bob?

Accepted Answer

$$v_{max} = \$$theta_0$$ \sqrt{gL}$$

Question 7

A basketball of mass $$M$$ and radius $$R is $$suspended by a wire and undergoes torsional oscillations with frequency $$f. $$What is the torsional constant $$\kappa of $$the wire?

Accepted Answer

$$\kappa = (2\pi f)^2$ \cdot \frac{2}{3} M R^2$

Question 8

Which of the following statements about the energy in simple harmonic motion is correct?

Accepted Answer

The total mechanical energy remains constant and is shared between kinetic and potential energy.

Question 9

A mass-spring system is driven at a frequency slightly above its natural frequency. Compared to the amplitude at resonance, the amplitude at this higher frequency is:

Accepted Answer

Smaller than the amplitude at resonance.

Question 10

A block of mass $$m is $$attached to a spring and oscillates with angular frequency $$\omega. If $$the mass is doubled and the spring constant remains the same, what happens to the angular frequency?

Accepted Answer

It decreases by a factor of $$\sqrt{2}.$$

Question 11

A physical pendulum consists of a uniform meter stick pivoted at one end. What is the period of small oscillations?

Accepted Answer

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

Question 12

A mass $$m$$ attached to a spring oscillates with amplitude $$A. At $$what displacement from equilibrium is the kinetic energy equal to the potential energy?

Accepted Answer

$$x = \frac{A}{\sqrt{2}}$$

Question 13

A car travels over bumps spaced $$d$$ meters apart. If the car's suspension has a natural frequency $$f_0$$, at what speed $$v$$ will the car experience resonance?

Accepted Answer

$$v = f_0$ d$$

Question 14

Which of the following is NOT a necessary condition for simple harmonic motion?

Accepted Answer

The restoring force is proportional to the square of the displacement.

Question 15

For a damped oscillator, which of the following best describes the effect of increasing the damping constant $$b$$?

Accepted Answer

The amplitude decays more quickly and the system returns to equilibrium faster.

Question 16

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

Accepted Answer

It is halved.

Question 17

Which of the following is the correct expression for the angular frequency $$\omega of a $$torsional pendulum with moment of inertia $$I$$ and torsional constant $$\kappa$$?

Accepted Answer

$$\omega = \sqrt{\frac{\kappa}{I}}$$

Question 18

A block attached to a spring is released from rest at $$x = +A. $$What is its velocity when it passes through the equilibrium position?

Accepted Answer

$$v = \omega A$$

Question 19

Which of the following best describes the phase constant $$\phi in $$the equation $$x(t) = A \cos(\omega t + \phi)$$ for simple harmonic motion?

Accepted Answer

It determines the initial position and direction of motion at $$t = 0.$$

Oscillations and Simple Harmonic Motion: Foundations for Physics with Calculus

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