Simple Harmonic Motion

rotational inertia

rotational inertia

I is the ________ of the hanging mass.

simple harmonic motion

A mass attached to the end of a spring will oscillate in ________.

Simple harmonic motion

________ is the study of oscillations.

simple harmonic motion

The pendulum is in ________ for small amplitudes.

A basketball player dribbles the ball so that it bounces regularly, twice per second. Is this ball in simple harmonic motion? Explain.

The ball is not in simple harmonic motion. An object in SHM experiences a force that pushes toward the center of the motion, pushing harder the farther the object is from the center; and, an object in SHM oscillates smoothly with a sinusoidal position–time graph. The basketball experiences only the gravitational force, except for the brief time that it’s in contact with the ground. Its position–time graph has sharp peaks when it hits the ground.

A pendulum has a period of 5 seconds on Earth. On Jupiter, where g ~ 30 m/s2 , the period of this pendulum would be closest to

(A) 1 s

(B) 3 s

(C) 5 s

(D) 8 s

(E) 15 s

B -

All that is changed by going to Jupiter is g, which is multiplied by 3. g is in the denominator and under a square root, so the period on Jupiter will be reduced by a factor of . So the original 5-second period is cut by a bit less than half, to about 3 seconds.

A mass on a spring has a frequency of 2.5 Hz and an amplitude of 0.05 m. In one complete period, what distance does the mass traverse? (This question asks for the actual distance, not the displacement.)

(A) 0.05 cm

(B) 0.01 cm

(C) 20 cm

(D) 10 cm

(E) 5 cm

C—

The amplitude of an object in SHM is the distance from equilibrium to the maximum displacement. In one full period, the mass traverses this distance four times: starting from max displacement, the mass goes down to the equilibrium position, down again to the max displacement on the opposite side, back to the equilibrium position, and back to where it started from. This is four amplitudes, or 0.20 m, or 20 cm.

Increasing which of the following will increase the period of a simple pendulum?

I. the length of the string

II. the local gravitational field

III. the mass attached to the string

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I, II, and III

A

Because L, the length of the string, is in the numerator, increasing L increases the period. Increasing g will actually decrease the period because g is in the denominator; increasing the mass on the pendulum has no effect because mass does not appear in the equation for period.

A mass m is attached to a horizontal spring of spring constant k. The spring oscillates in simple harmonic motion with amplitude A.

At what displacement from equilibrium is the speed half of the maximum value?

The maximum speed of the mass is at the equilibrium position, where PE = 0, so all energy is kinetic. The maximum potential energy is at the maximum displacement A, because there the mass is at rest briefly and so has no KE.

Use conservation of energy to equate the energy of the maximum displacement and position 2.

86% of the amplitude.

A mass m is attached to a horizontal spring of spring constant k. The spring oscillates in simple harmonic motion with amplitude A.

At what displacement from equilibrium is the potential energy half of the maximum value?

At some position x, the potential energy will be 1/2 of its maximum value. This works out to about 70% of the maximum amplitude.