Question 1

In the context of damped oscillations, which curve (A, B, or C) in the first image best represents an underdamped oscillator, and why?

Accepted Answer

Curve C, because it oscillates with decreasing amplitude and takes the longest to return to equilibrium.

Question 2

A car shock absorber, as shown in the second image, is designed to provide which type of damping for the car's suspension system?

Accepted Answer

Critical damping, so the car returns to equilibrium as quickly as possible without oscillating.

Question 3

The energy of an underdamped oscillator decreases over time according to which of the following equations?

Accepted Answer

$$E = E_0$ e$$^{-\frac{b}$${m} t}$$

Question 4

A guitar string, as shown in the third image, loses half its energy in 2.25 s. If its fundamental frequency is 82.41 Hz, what is the quality factor $$Q of $$the string? (Use $$Q = 2 \pi f 	au$$ and $$	au = \frac{2.25}{\ln 2})$$

Accepted Answer

$$Q \approx 168$$

Question 5

Which of the following best describes the relationship between the quality factor $$Q$$ and the width $$\Delta \omega of $$the resonance curve for a weakly damped oscillator?

Accepted Answer

$$Q = \$$omega_0$$ / \Delta \omega$$

Question 6

For a mass-spring system with mass $$m$$ and spring constant $$k$$, what is the condition for critical damping in terms of the damping constant $$b$$?

Accepted Answer

$$b = 2 \sqrt{km}$$

Question 7

If a damped oscillator is driven by an external force $$F_{	ext{external}} = F_0$ \cos(\omega t)$$, what is the amplitude $$A of $$the steady-state oscillation?

Accepted Answer

$$A = \frac{F_0$}{m \sqrt{(\$$omega_0$$^2 - \$$omega^2$$)^2 + (b$$\$$omega/m)^2$$}}$

Question 8

Which of the following statements about resonance in a forced oscillator is correct?

Accepted Answer

Resonance occurs when the driving frequency equals the natural frequency, resulting in maximum amplitude.

Question 9

A system is said to be overdamped if:

Accepted Answer

$$b \u003e 2\sqrt{km}$$

Question 10

For a weakly damped oscillator, which of the following is true about the quality factor $$Q$$ and the energy loss per period?

Accepted Answer

A large $$Q$$ means a small energy loss per period.

Question 11

Given the equation for the damped frequency $$\omega' = \sqrt{\$$omega_0$$^2 - (b/2m)^2$}$$, what happens to $$\omega' as $$the damping constant $$b$$ increases?

Accepted Answer

$$\omega'$$ decreases.

Question 12

If the energy of a damped oscillator decays to $$1/e of $$its initial value after a time $$	au$$, what is the relationship between $$	au$$, $$m$$, and $$b$$?

Accepted Answer

$$	au = m/b$$

Question 13

Which of the following best describes the phase relationship between the displacement and the driving force at resonance?

Accepted Answer

They are out of phase by $$90^\circ$$ ($$\phi = \pi/2).$$

Question 14

A mass-spring system with $$m = 0.155 kg $$and $$k = 525 N/m is $$driven at resonance. If the Q factor is 126, what is the width of the resonance curve $$\Delta \omega$$?

Accepted Answer

$$\Delta \omega = 0.462$$ rad/s

Question 15

Which of the following is a correct expression for the quality factor $$Q of a $$damped oscillator?

Accepted Answer

$$Q = \frac{m \$$omega_0$$}{b}$$

Question 16

If a system is critically damped, what is the behavior of its displacement after being displaced from equilibrium?

Accepted Answer

It returns to equilibrium without oscillating, in the shortest possible time.

Question 17

For a forced oscillator, what happens to the amplitude as the driving frequency moves far away from the natural frequency?

Accepted Answer

The amplitude decreases.

Question 18

Which of the following best describes the effect of increasing the damping constant $$b on $$the resonance curve of a forced oscillator?

Accepted Answer

The resonance peak becomes broader and lower.

Question 19

A weakly damped oscillator has a Q factor of 500. What is the approximate fractional energy loss per period?

Accepted Answer

$$\frac{\Delta E}{E} \approx \frac{2\pi}{Q} = 0.0126$$

Damped and Forced Oscillations: Simple Harmonic Motion, Resonance, and Quality Factor

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