Q: A mass of $$8\ \mathrm{kg}$$ stretches a spring by $$3.6\ \mathrm{cm} at $$equilibrium. What is the spring constant $$k of $$the spring?

$$k = \frac{mg}{\Delta x} = \frac{8 \times 9.8}{0.036} = 2178\ \mathrm{N/m}$$

Question 1

A block of mass $$m is $$attached to a spring with force constant $$k$$ and oscillates on a frictionless surface. If the amplitude of oscillation is $$A$$, what is the expression for the maximum speed of the block during its motion?

Accepted Answer

$$v_{max} = A \sqrt{\frac{k}{m}}$$

Question 2

A simple pendulum has a length $$L$$ and swings with a small amplitude. Which of the following correctly gives the period $$T of $$the pendulum?

Accepted Answer

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

Question 3

A mass of $$8\ \mathrm{kg}$$ stretches a spring by $$3.6\ \mathrm{cm} at $$equilibrium. What is the spring constant $$k of $$the spring?

Accepted Answer

$$k = \frac{mg}{\Delta x} = \frac{8 	imes 9.8}{0.036} = 2178\ \mathrm{N/m}$$

Question 4

Which of the following statements about simple harmonic motion (SHM) is correct?

Accepted Answer

The restoring force is always directed toward the equilibrium position and is proportional to the displacement.

Question 5

A mass attached to a vertical spring oscillates up and down. At the instant it passes through the equilibrium position moving upward, what are the signs of its velocity and acceleration (taking upward as positive)?

Accepted Answer

Velocity is positive, acceleration is negative.

Question 6

The total mechanical energy $$E of a $$mass-spring system in SHM is given by which of the following expressions?

Accepted Answer

$$E = \frac{1}{2}kA^2$

Question 7

A simple pendulum on Earth has a period of $$1.75\ \mathrm{s}. On $$Planet X, the same pendulum has a period of $$2.14\ \mathrm{s}. $$What is the acceleration due to gravity on Planet X?

Accepted Answer

$$g_X = g_{Earth} \left(\frac{T_{Earth}}{T_X}\$$right)^2$$ = 9.8 \left(\frac{1.75}{2.14}\$$right)^2$$ = 6.55\ \mathrm{m/s^2$}$$

Question 8

Which of the following best describes the effect of increasing the phase angle $$\phi in $$the equation $$x = A \cos(\omega t + \phi)$$ for SHM?

Accepted Answer

It shifts the displacement vs. time graph to the left.

Question 9

A $$2.45\ \mathrm{N}$$ object is attached to a spring and oscillates with a period of $$0.64\ \mathrm{s}. $$What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{4\pi^2$ m}{T^2$} = \frac{4\pi^2$ (2.45/9.8)}{(0.64)^2$} = 24\ \mathrm{N/m}$$

Question 10

Which of the following is true for the energy in a simple harmonic oscillator (ignoring friction)?

Accepted Answer

The total mechanical energy remains constant, but potential and kinetic energy transform into each other.

Question 11

A child on a swing (modeled as a simple pendulum) increases the amplitude of oscillation by pumping their legs. What happens to the period of the swing if the amplitude remains small?

Accepted Answer

The period remains approximately the same.

Question 12

Which of the following equations correctly gives the angular frequency $$\omega$$ for a mass-spring system in SHM?

Accepted Answer

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

Question 13

A mass-spring system oscillates with amplitude $$A$$ and period $$T. At $$what displacement is the kinetic energy equal to the potential energy?

Accepted Answer

$$x = \pm \frac{A}{\sqrt{2}}$$

Question 14

Which of the following is a necessary condition for the motion of a simple pendulum to be considered simple harmonic?

Accepted Answer

The angular displacement $$	heta$$ must be small so that $$\sin 	heta \approx 	heta.$$

Question 15

A block attached to a spring is released from rest at $$x = A. $$What is the velocity of the block when it passes through the equilibrium position ($$x = 0$$)?

Accepted Answer

$$v = \omega A$$

Question 16

If the frequency of a mass-spring system is doubled, what happens to the period?

Accepted Answer

The period is halved.

Question 17

A mass $$m$$ oscillates on a spring with force constant $$k. If $$the mass is increased to $$4m$$, what happens to the period of oscillation?

Accepted Answer

The period doubles.

Question 18

Which of the following best describes the relationship between the restoring force $$F_x$$ and displacement $$x$$ for a mass-spring system?

Accepted Answer

$$F_x = -kx$$

Question 19

A grandfather clock uses a pendulum to keep time. If the clock is moved to a location where $$g is $$smaller, what happens to the period of the pendulum?

Accepted Answer

The period increases.

Simple Harmonic Motion and the Simple Pendulum

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