Q: The quality factor $$Q of a $$damped oscillator is defined as:

$$Q = \frac{\$$omega_0$$ \tau}{1}$$

Question 1

A guitar string is plucked and loses half its energy after 2.25 s. If its fundamental frequency is 82.41 Hz, what is the decay time $$	au of $$the string?

Accepted Answer

$$	au = \frac{2.25}{\ln 2}$$

Question 2

Which equation best describes the amplitude $$A_0 of a $$forced, damped oscillator driven at angular frequency $$\omega$$?

Accepted Answer

$$A_0$$ = \frac{$$F_0$$}{m\left(\$$omega_0^2$$ - \$$omega^2$$\right) + $$b^2$$ \$$omega^2$$}$$

Question 3

The quality factor $$Q of a $$damped oscillator is defined as:

Accepted Answer

$$Q = \frac{\$$omega_0$$ 	au}{1}$$

Question 4

For a weakly damped oscillator, what happens to the resonance curve as the damping increases?

Accepted Answer

The curve broadens and becomes shorter.

Question 5

If the energy loss per period is 5%, what is the quality factor $$Q$$?

Accepted Answer

$$Q = \frac{2\pi}{0.05}$$

Question 6

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

Accepted Answer

$$Q = \frac{\$$omega_0$$}{\Delta \omega}$$

Question 7

A mass-spring system is driven at its natural frequency. What is the phase difference $$\$$phi_0$$ between the driving force and the displacement?

Accepted Answer

$$\$$phi_0$$ = 90^\circ$$

Question 8

The energy of an underdamped oscillator decreases exponentially with time. Which equation describes this behavior?

Accepted Answer

$$E(t) = E_0$ e$$^{-t/	au}$$

Question 9

In the context of forced oscillations, what happens to the amplitude as the driving frequency approaches the natural frequency?

Accepted Answer

The amplitude increases and reaches a maximum at resonance.

Question 10

For a damped oscillator, the fractional energy loss per period is given by:

Accepted Answer

$$\frac{\Delta E_{period}}{E_0$} = \frac{2\pi}{Q}$$

Question 11

A 155 g mass oscillates on a spring with $$k = 525 N/m. $$What is its natural angular frequency $$\$$omega_0$$?

Accepted Answer

$$\$$omega_0$$ = \sqrt{\frac{k}{m}}$$

Question 12

Which of the following statements about the quality factor $$Q is $$correct?

Accepted Answer

A high $$Q$$ means the system is weakly damped and loses little energy per cycle.

Question 13

If a system is driven at a frequency much lower than its natural frequency, what is the phase difference between the driving force and the displacement?

Accepted Answer

$$\$$phi_0$$ = 0^\circ$$

Question 14

The equation $$m \frac{d^2$ s}{dt^2$} + b \frac{ds}{dt} + k s = F_0$ \cos \omega t$$ describes:

Accepted Answer

A forced, damped oscillator

Question 15

At resonance, the amplitude of a forced oscillator is maximized. Which parameter is most responsible for the sharpness of the resonance peak?

Accepted Answer

The quality factor $$Q$$

Question 16

For a weakly damped oscillator, the damped frequency $$\omega' is $$related to the undamped frequency $$\$$omega_0$$ and Q$ by:

Accepted Answer

$$\omega' = \$$omega_0$$ \left(1 - \frac{1}{2Q^2$}\right)$$

Question 17

A system with a large $$Q$$ factor will have which of the following characteristics?

Accepted Answer

Sharp resonance curve and low energy loss per cycle

Question 18

If the amplitude at resonance is $$A_0$$ = \frac{$$F_0$$}{b \$$omega_0$$}$$, what happens to A_0$ if the damping coefficient b$ increases?

Accepted Answer

The amplitude decreases.

Question 19

Which physical scenario best illustrates resonance in a real-world context?

Accepted Answer

A guitar string vibrating when plucked and then being driven by a speaker at its natural frequency.

Oscillations, Damping, and Resonance: Quality Factor and Forced Oscillations

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