Q: The position of a mass on a spring is given by $$x(t) = 0.2 \sin(5t + \dfrac{\pi}{3}) m. $$What is the maximum velocity of the mass?

$$v_{\text{max}} = 0.2 \times 5 = 1.0 m/s$$

Q: For a mass-spring system in SHM, which of the following is the correct expression for the total mechanical energy?

$$E_{\text{mec}} = \dfrac{1}{2} k A^2$

Question 1

A mass $$m is $$attached to a horizontal spring with spring constant $$k$$ and oscillates with no friction. What is the correct expression for the angular frequency $$\omega of $$the oscillation?

Accepted Answer

$$\omega = \sqrt{\dfrac{k}{m}}$$

Question 2

Which of the following statements about simple harmonic motion (SHM) is TRUE?

Accepted Answer

The acceleration in SHM is always opposite in direction to the displacement from equilibrium.

Question 3

A 0.5 kg mass is attached to a spring with $$k = 8 N/m. $$What is the period $$T of $$oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\dfrac{0.5}{8}} \approx 1.57 s$$

Question 4

The position of a mass on a spring is given by $$x(t) = 0.2 \sin(5t + \dfrac{\pi}{3}) m. $$What is the maximum velocity of the mass?

Accepted Answer

$$v_{	ext{max}} = 0.2 	imes 5 = 1.0 m/s$$

Question 5

A simple pendulum of length $$L$$ swings with small amplitude. Which of the following correctly gives its angular frequency $$\omega$$?

Accepted Answer

$$\omega = \sqrt{\dfrac{g}{L}}$$

Question 6

For a mass-spring system in SHM, which of the following is the correct expression for the total mechanical energy?

Accepted Answer

$$E_{	ext{mec}} = \dfrac{1}{2} k A^2$

Question 7

A mass is oscillating on a spring with amplitude $$A$$ and angular frequency $$\omega. At $$what position is the kinetic energy maximized?

Accepted Answer

At $$x = 0$$, the equilibrium position

Question 8

Which of the following best describes the relationship between simple harmonic motion and uniform circular motion?

Accepted Answer

The projection of uniform circular motion onto one axis is simple harmonic motion.

Question 9

A simple pendulum has a length of $$1.0 m. $$What is its period $$T ($$take $$g = 9.8 m/s$^2$$)?

Accepted Answer

$$T = 2\pi \sqrt{\dfrac{1.0}{9.8}} \approx 2.01 s$$

Question 10

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

Accepted Answer

The period increases by a factor of $$\sqrt{2}$$

Question 11

Which of the following is a correct statement about the energy transformation in a simple harmonic oscillator?

Accepted Answer

Potential energy is maximum when the displacement is maximum.

Question 12

A mass on a spring is described by $$x(t) = 0.1 \sin(7.85 t - \dfrac{\pi}{2}). $$What is the phase constant $$\phi in $$radians?

Accepted Answer

$$-\dfrac{\pi}{2}$$

Question 13

For a simple pendulum, which of the following approximations is valid for small angles $$	heta$$?

Accepted Answer

$$\sin 	heta \approx 	heta$$

Question 14

A hydrogen molecule can be modeled as two atoms connected by a spring. For small oscillations, what is the form of the potential energy near equilibrium?

Accepted Answer

$$U(x) = \dfrac{1}{2} k (x - x_0$)^2$

Question 15

If the equation of motion for a mass-spring system is $$x(t) = A \cos(\omega t + \phi)$$, what is the corresponding velocity equation?

Accepted Answer

$$v(t) = -\omega A \sin(\omega t + \phi)$$

Question 16

A mass $$m$$ oscillates on a spring with amplitude $$A$$ and angular frequency $$\omega. $$What is the maximum acceleration?

Accepted Answer

$$a_{	ext{max}} = \$$omega^2$$ A$$

Question 17

Which of the following is NOT an assumption for a simple pendulum to exhibit simple harmonic motion?

Accepted Answer

The amplitude of oscillation is large.

Question 18

If the total mechanical energy of a mass-spring system is $$E_{	ext{mec}} = 2 J $$and the amplitude is $$A = 0.5 m$$, what is the spring constant $$k$$?

Accepted Answer

$$k = \dfrac{2 	imes 2}{(0.5)^2$} = 16 N/m$$

Question 19

A mass on a spring is released from rest at $$x = A. $$What is the phase constant $$\phi if $$the position is described by $$x(t) = A \sin(\omega t + \phi)$$?

Accepted Answer

$$\phi = \dfrac{\pi}{2}$$

Chapter 14: Oscillations – Simple Harmonic Motion, Springs, and Pendulums

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