Q: A block of mass 4.0 kg attached to a horizontal spring oscillates with amplitude 0.50 m and maximum acceleration 20 m/s$$^2. $$What is the spring constant $$k$$?

$$k = \frac{ma_{max}}{A} = \frac{4.0 \times 20}{0.50} = 160 N/m$$

Q: A block is hung from an ideal spring with force constant $$k = 40 N/m $$and stretches 0.120 m at equilibrium. What is the mass of the block?

$$m = \frac{kx}{g} = \frac{40 \times 0.120}{9.8} \approx 0.49 kg$$

Question 1

Which of the following best describes simple harmonic motion (SHM)?

Accepted Answer

A motion that repeats in a set time period, with a restoring force proportional to the displacement and directed toward equilibrium.

Question 2

A 0.245 kg mass is attached to a spring with a spring constant of 7580 N/m and placed on a frictionless surface. What is the frequency of oscillation?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{7580}{0.245}} \approx 28 Hz$$

Question 3

Which equation correctly gives the angular frequency $$\omega$$ for a mass $$m on a $$spring with spring constant $$k$$?

Accepted Answer

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

Question 4

A block of mass 4.0 kg attached to a horizontal spring oscillates with amplitude 0.50 m and maximum acceleration 20 m/s$$^2. $$What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{ma_{max}}{A} = \frac{4.0 	imes 20}{0.50} = 160 N/m$$

Question 5

If a mass-spring system oscillates with period $$T$$, what is the relationship between period and mass?

Accepted Answer

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

Question 6

A block is hung from an ideal spring with force constant $$k = 40 N/m $$and stretches 0.120 m at equilibrium. What is the mass of the block?

Accepted Answer

$$m = \frac{kx}{g} = \frac{40 	imes 0.120}{9.8} \approx 0.49 kg$$

Question 7

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

Accepted Answer

The total mechanical energy is constant and is the sum of kinetic and potential energies.

Question 8

At what position during SHM is the speed of the oscillating object maximum?

Accepted Answer

At the equilibrium position ($$x = 0)$$

Question 9

A mass-spring system oscillates with amplitude $$A. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2}kA^2$

Question 10

If the period of a mass-spring system on Earth is $$T$$, what will be the period if the same system is placed on the Moon (ignoring air resistance)?

Accepted Answer

The period will remain the same because $$T$$ depends only on $$m$$ and $$k.$$

Question 11

A mass $$m$$ attached to a spring oscillates with displacement $$x(t) = A \cos(\omega t). $$What is the velocity as a function of time?

Accepted Answer

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

Question 12

A 0.5 kg mass oscillates on a spring with $$k = 200 N/m. $$What is the period of oscillation?

Accepted Answer

$$T = 2\pi \sqrt{\frac{0.5}{200}} \approx 0.314 s$$

Question 13

Which of the following is true about the frequency $$f$$ and period $$T of a $$simple harmonic oscillator?

Accepted Answer

$$f = \frac{1}{T}$$

Question 14

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

Accepted Answer

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

Question 15

A 1.0 kg mass is attached to a spring and oscillates with a period of 2.0 s. What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ 	imes 1.0}{4} \approx 9.87 N/m$$

Question 16

Which of the following real-world systems can be best modeled as a simple harmonic oscillator?

Accepted Answer

A child swinging on a small-angle playground swing

Question 17

If the amplitude of a mass-spring oscillator is doubled, what happens to the total mechanical energy?

Accepted Answer

It increases by a factor of 4

Question 18

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 maximum acceleration?

Accepted Answer

$$a_{max} = \$$omega^2$$ A = 5^2$ 	imes 0.10 = 2.5 m/s$^2$$

Question 19

Which of the following best explains why the motion of a mass on a spring is sinusoidal?

Accepted Answer

Because the restoring force is proportional to displacement, leading to a second-order differential equation with sinusoidal solutions.

Chapter 11: Periodic Motion and Simple Harmonic Motion (SHM)

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