Small angle approximation ($$\theta \ll 1$$ rad)

Question 1

Which of the following best describes the restoring force in a simple pendulum for small angles?

Accepted Answer

The restoring force is provided by gravity, and for small angles, it is proportional to the angle $$	heta.$$

Question 2

For a simple pendulum, what is the expression for the angular frequency $$\omega$$?

Accepted Answer

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

Question 3

A pendulum is displaced by a small angle and released. Which approximation allows us to treat its motion as simple harmonic?

Accepted Answer

Small angle approximation ($$	heta \ll 1$$ rad)

Question 4

In the context of damped harmonic motion, what does 'critical damping' refer to?

Accepted Answer

The minimum amount of damping needed to eliminate oscillations and return to equilibrium as quickly as possible.

Question 5

Which scenario best illustrates resonance?

Accepted Answer

A playground swing being pushed at its natural frequency, causing the amplitude to increase.

Question 6

Given a simple pendulum with length $$L = 2 m $$and $$g = 9.8 m/s$^2$$, what is its angular frequency $$\omega$$?

Accepted Answer

$$\omega = \sqrt{\frac{9.8}{2}} \approx 2.21$$ rad/s

Question 7

Which of the following best describes overdamped motion?

Accepted Answer

The system returns to equilibrium without oscillating, slower than the critically damped case.

Question 8

What is the moment of inertia $$I$$ for a simple pendulum with mass $$m$$ and length $$L$$?

Accepted Answer

$$I = mL^2$

Question 9

Which equation represents the natural frequency $$f of a $$mass-spring system?

Accepted Answer

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

Question 10

A damped harmonic oscillator loses energy due to friction. Which of the following best describes its amplitude over time?

Accepted Answer

The amplitude decreases over time.

Question 11

In the context of orbital resonance, as seen in Jupiter's moons, what does a 4:1 resonance mean?

Accepted Answer

The inner moon completes 4 orbits for every 1 orbit of the outer moon.

Question 12

Which of the following is NOT a source of damping in harmonic motion?

Accepted Answer

Gravity

Question 13

A mass-spring system is driven at its natural frequency. What phenomenon is most likely to occur?

Accepted Answer

Resonance

Question 14

Which of the following best describes the effect of resonance in the Tacoma Narrows Bridge collapse?

Accepted Answer

Resonance caused the bridge to oscillate at its natural frequency, leading to destructive amplitude increases.

Question 15

If a pendulum is released from a small angle, which equation describes the torque acting on the bob?

Accepted Answer

$$	au = -mgL	heta$$

Question 16

Which of the following is a real-life example of driven harmonic motion?

Accepted Answer

A child pumping their legs to increase the amplitude of a playground swing.

Question 17

What happens to the amplitude of a driven harmonic oscillator when the driving frequency matches the natural frequency?

Accepted Answer

The amplitude increases significantly (resonance).

Question 18

Which of the following best describes underdamped motion?

Accepted Answer

The system oscillates with decreasing amplitude.

Question 19

Which equation relates the arc length $$s to $$the angle $$	heta$$ and length $$L of a $$pendulum?

Accepted Answer

$$s = L	heta$$

Question 20

Which of the following is a beneficial application of resonance?

Accepted Answer

Musical instruments amplifying sound at specific frequencies.

Simple Harmonic Motion: Pendulums, Damping, and Resonance- lecture 24

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