Q: For a simple pendulum, the restoring force is proportional to $$\sin \theta. $$For small angles, why can we approximate $$\sin \theta \approx \theta$$?

Because for small $$\theta$$, the difference is negligible.

Question 1

Which equation describes the restoring force for a mass attached to an ideal spring according to Hooke's Law?

Accepted Answer

$$F_x = -kx$$

Question 2

A block of mass $$m$$ attached to a spring with force constant $$k$$ oscillates with angular frequency $$\omega. $$Which formula gives $$\omega$$ for this system?

Accepted Answer

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

Question 3

The displacement $$x of a $$simple harmonic oscillator as a function of time is given by $$x = A \cos(\omega t + \phi). $$What does $$A$$ represent?

Accepted Answer

The amplitude of oscillation

Question 4

A pendulum has a length $$L$$ and swings with small angles. Which equation best approximates its period $$T$$?

Accepted Answer

$$T = 2\pi \sqrt{\frac{L}{g}}$$

Question 5

If the amplitude of a damped oscillator decreases from $$A = 2 cm at$$ $$t = 0 to$$ $$A = 1 cm at$$ $$t = 15 s$$, and the damping constant is $$b = 0.1 kg/s$$, what is the mass $$m of $$the oscillating object? (Use $$A = A_0$ e$$^{-\frac{b}$${2m}t})$$

Accepted Answer

$$m = 0.22 kg$$

Question 6

Which of the following best describes the energy in a simple harmonic oscillator?

Accepted Answer

Total mechanical energy is conserved because only conservative forces act.

Question 7

For a damped oscillator, the angular frequency $$\omega' is $$given by:

Accepted Answer

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

Question 8

A building in an earthquake-prone area should have a natural frequency that is:

Accepted Answer

Very different from typical earthquake frequencies.

Question 9

Which equation gives the period $$T of a $$mass-spring system in simple harmonic motion?

Accepted Answer

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

Question 10

In a damped oscillator, what happens to the amplitude over time?

Accepted Answer

It decreases exponentially.

Question 11

A mass $$m on a $$spring is displaced to $$x = A$$ and released. At $$x = 0$$, what is true about its velocity and acceleration?

Accepted Answer

Velocity is maximum, acceleration is zero.

Question 12

For a simple pendulum, the restoring force is proportional to $$\sin 	heta. $$For small angles, why can we approximate $$\sin 	heta \approx 	heta$$?

Accepted Answer

Because for small $$	heta$$, the difference is negligible.

Question 13

Which of the following best describes resonance in forced oscillations?

Accepted Answer

Resonance occurs when the driving frequency is approximately equal to the natural frequency.

Question 14

A mass-spring system is oscillating vertically. If the spring constant $$k is $$doubled, what happens to the period $$T$$?

Accepted Answer

The period decreases by a factor of $$\sqrt{2}.$$

Question 15

In SHM, the total mechanical energy is given by $$E = \frac{1}{2}kA^2$. What does A$ represent?

Accepted Answer

The maximum displacement (amplitude)

Question 16

A mass $$m$$ attached to a spring oscillates with frequency $$f. $$Which equation gives $$f$$?

Accepted Answer

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

Question 17

A pendulum on Earth has a period of $$1.75 s. On $$Planet X, its period is $$2.14 s. $$What is the acceleration due to gravity on Planet X? (Assume length is unchanged.)

Accepted Answer

$$g = 6.55 m/s$^2$$

Question 18

Which force acts as the damping force in a damped oscillator?

Accepted Answer

$$F_D = -b v_x$$

Question 19

In a damped oscillator, what effect does increasing the damping constant $$b$$ have on the period $$T$$?

Accepted Answer

The period increases.

Question 20

A mass-spring system is subjected to a periodic driving force with frequency $$\omega_d. $$What happens if $$\omega_d is $$close to the system's natural frequency $$\omega'$$?

Accepted Answer

The amplitude of oscillation increases dramatically (resonance).

Simple Harmonic Motion, Damped Oscillations, and Resonance

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