Q: A 132 kg mass oscillates in simple harmonic motion (SHM) with an amplitude of 5.00 cm. If the spring constant is $$k = 1.20 \times 10^5$ \ \text{kg/s}^2$$, what is the natural frequency of the system?

$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = 4.80 \ \text{Hz}$$

Q: An astronaut observes that a 1.00 m pendulum completes 100 cycles in 496 seconds. What is the acceleration due to gravity at the astronaut’s location?

$$g = \frac{4\pi^2$ l}{T^2$} = 1.60 \ \text{m/s}^2$$

Q: If the amplitude of a damped oscillator decreases to $$\frac{1}{3}$$ its original value after 20 oscillations at a frequency of 2.00 Hz, what is the value of $$\gamma$$?

$$\gamma = \frac{1}{10} \ln 3 = 0.11 \ \text{s}^{-1}$$

Q: A load on a spring vibrates at a natural frequency of 4.00 Hz and initial amplitude of 20.0 cm. After 10 oscillations, the amplitude is halved. What is the amplitude after 1.25 periods?

$$A = 20.0 \times e$$^{-0.277 \times 0.3125}$$ = 18.3 \ \text{cm}$$

Question 1

A 132 kg mass oscillates in simple harmonic motion (SHM) with an amplitude of 5.00 cm. If the spring constant is $$k = 1.20 	imes 10^5$ \ 	ext{kg/s}^2$$, what is the natural frequency of the system?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = 4.80 \ 	ext{Hz}$$

Question 2

For a simple pendulum of length $$l$$ and mass $$m$$, what is the expression for the restoring force when the angle $$	heta is $$small?

Accepted Answer

$$F_R = -mg	heta$$

Question 3

Which approximation is valid for small angles ($$	heta \u003c 15^\circ) in $$the context of a simple pendulum?

Accepted Answer

$$\sin	heta \approx 	heta$$

Question 4

An astronaut observes that a 1.00 m pendulum completes 100 cycles in 496 seconds. What is the acceleration due to gravity at the astronaut’s location?

Accepted Answer

$$g = \frac{4\pi^2$ l}{T^2$} = 1.60 \ 	ext{m/s}^2$$

Question 5

What is the general solution for the displacement $$s(t) of a $$damped harmonic oscillator?

Accepted Answer

$$s(t) = A_0$ e$$^{-\gamma t}$$ \cos(\omega' t)$$

Question 6

In the context of damped oscillations, what does the parameter $$\gamma$$ represent?

Accepted Answer

$$\gamma = \frac{b}{2m}$$, the damping coefficient

Question 7

A damped oscillator has a spring constant $$k$$, mass $$m$$, and damping constant $$b. $$What is the expression for the angular frequency $$\omega' of $$the damped system?

Accepted Answer

$$\omega' = \sqrt{\frac{k}{m} - \frac{b^2$}{4m^2$}}$$

Question 8

If the amplitude of a damped oscillator decreases to $$\frac{1}{3}$$ its original value after 20 oscillations at a frequency of 2.00 Hz, what is the value of $$\gamma$$?

Accepted Answer

$$\gamma = \frac{1}{10} \ln 3 = 0.11 \ 	ext{s}^{-1}$$

Question 9

A load on a spring vibrates at a natural frequency of 4.00 Hz and initial amplitude of 20.0 cm. After 10 oscillations, the amplitude is halved. What is the amplitude after 1.25 periods?

Accepted Answer

$$A = 20.0 	imes e$$^{-0.277 	imes 0.3125}$$ = 18.3 \ 	ext{cm}$$

Question 10

Which equation describes the period $$T of a $$mass-spring system in SHM?

Accepted Answer

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

Question 11

For a simple pendulum, what is the expression for its natural frequency?

Accepted Answer

$$f = \frac{1}{2\pi} \sqrt{\frac{g}{l}}$$

Question 12

A damped oscillator’s amplitude decays exponentially. What is the mathematical form of this decay?

Accepted Answer

$$A(t) = A_0$ e$$^{-\gamma t}$$

Question 13

Which force is responsible for damping in a real-world oscillator, such as a sphere falling through a viscous liquid?

Accepted Answer

$$F_d = -bv$$

Question 14

If a pendulum is taken to the Moon, where $$g_{moon} \approx \frac{1}{6} g_{earth}$$, how does its period compare to its period on Earth?

Accepted Answer

The period increases by a factor of $$\sqrt{6}$$

Question 15

Which of the following best describes the motion shown in the second image, where amplitude decreases over time?

Accepted Answer

Damped harmonic oscillations with exponential decay

Question 16

A mass-spring system is subject to a damping force proportional to velocity. Which equation correctly represents the net force?

Accepted Answer

$$F = -ks - bv$$

Question 17

If the displacement of a damped oscillator is given by $$s(t) = (5.45 \ 	ext{cm}) e$$^{-4.00 t}$$ \cos(9.42 t)$$, what is the frequency of oscillation?

Accepted Answer

$$f = \frac{9.42}{2\pi} = 1.50 \ 	ext{Hz}$$

Question 18

Which Taylor series expansion is used to justify the small angle approximation for $$\sin	heta$$?

Accepted Answer

$$\sin	heta = 	heta - \frac{\$$theta^3$$}{3!} + \frac{\$$theta^5$$}{5!} - \cdots$$

Simple Harmonic Motion and Damped Oscillations: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards