Q: A 0.5 kg mass attached to a spring oscillates with a period of 2 s. What is the spring constant $$k$$?

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ \times 0.5}{4} = \pi^2$ N/m

Question 1

A mass $$m is $$attached to a spring with spring constant $$k$$ and oscillates with simple harmonic motion. Which of the following expressions gives the period $$T of $$the oscillation?

Accepted Answer

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

Question 2

A simple pendulum of length $$L$$ swings with small amplitude. Which of the following correctly gives its period $$T$$?

Accepted Answer

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

Question 3

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

Accepted Answer

The restoring force is always proportional to the displacement from equilibrium.

Question 4

A mass on a spring oscillates with amplitude $$A$$ and angular frequency $$\omega. $$What is the maximum speed $$v_{max} of $$the mass?

Accepted Answer

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

Question 5

The total mechanical energy $$E of a $$mass-spring system in SHM is given by which of the following expressions?

Accepted Answer

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

Question 6

A 0.5 kg mass attached to a spring oscillates with a period of 2 s. What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ 	imes 0.5}{4} = \pi^2$ N/m

Question 7

Which of the following best describes the effect of damping on a harmonic oscillator?

Accepted Answer

The amplitude decreases exponentially over time.

Question 8

A mass-spring system is described by $$x(t) = 0.10 \cos(5t)$$, where $$x is in $$meters and $$t in $$seconds. What is the frequency $$f of $$the oscillation?

Accepted Answer

$$f = \frac{5}{2\pi} Hz$$

Question 9

Which of the following is true for the period of a simple pendulum?

Accepted Answer

It depends only on the length of the pendulum and the acceleration due to gravity.

Question 10

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 11

A mass $$m$$ attached to a spring is displaced from equilibrium and released from rest. Which of the following initial conditions is correct?

Accepted Answer

$$x(0) = A$$, $$v(0) = 0$$

Question 12

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

Accepted Answer

It occurs when the driving frequency matches the natural frequency, resulting in large amplitude oscillations.

Question 13

A damped oscillator is described by $$x(t) = A e$$^{-\gamma t}$$ \cos(\omega t + \phi). $$What does the parameter $$\gamma$$ represent?

Accepted Answer

The rate at which the amplitude decays.

Question 14

A mass-spring system has a total mechanical energy $$E. At $$what displacement $$x is $$the kinetic energy equal to the potential energy?

Accepted Answer

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

Question 15

Which of the following is NOT a characteristic of simple harmonic motion?

Accepted Answer

The period depends on amplitude.

Question 16

A mass $$m$$ oscillates on a spring with amplitude $$A$$ and angular frequency $$\omega. $$What is the expression for its acceleration as a function of time?

Accepted Answer

$$a(t) = -\$$omega^2$$ x(t)$$

Question 17

A car's suspension system can be modeled as a damped harmonic oscillator. What is the practical purpose of damping in this context?

Accepted Answer

To reduce the amplitude of oscillations and return the car to equilibrium smoothly.

Question 18

Which of the following is the correct relationship between frequency $$f$$, period $$T$$, and angular frequency $$\omega$$ for SHM?

Accepted Answer

$$f = \frac{1}{T}$$ and $$\omega = 2\pi f$$

Question 19

A molecule can be modeled as two atoms connected by a spring. If the equilibrium separation is $$R_0$$ and the potential energy minimum is $$U_0$$, what is the effective spring constant $$k$$ for small oscillations?

Accepted Answer

$$k = \frac{72 U_0$}{R_0$^2}$$

Chapter 14: Periodic Motion and Simple Harmonic Oscillators

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