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Simple Harmonic Motion

- Humans are machines for identifying patterns, it’s fundamental to how our brains work.
- SHM is a repeating pattern in motion, describes many patterns we see in nature.
- Covered at a basic level on the AP test, but has lots of vocabulary.

Oscillation

- Medium
- Disturbance
- Restoring force
- Examples:
- Ocean waves
- Pendulum
- Spring

Period, Frequency

- Period (T): The time to complete one cycle.
- Frequency (f): The number of cycles in one second.

Practice Problem

- A girl in boat watches waves on a lake passing with a half-second pause between each crest. She notices that each wave takes 1.5 s to sweep straight down the length of her 4.5 m boat.
- Determine
- a) the period of the waves
- b) the frequency of the waves

ConcepTest 11.1aHarmonic Motion I

1) 0

2) A/2

3) A

4) 2A

5) 4A

A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T?

ConcepTest 11.1aHarmonic Motion I

1) 0

2) A/2

3) A

4) 2A

5) 4A

A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T?

In the time interval T (the period), the mass goes through one complete oscillation back to the starting point. The distance it covers is: A + A + A + A (4A).

ConcepTest 11.1bHarmonic Motion II

1) 0

2) A/2

3) A

4) 2A

5) 4A

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T?

ConcepTest 11.1bHarmonic Motion II

1) 0

2) A/2

3) A

4) 2A

5) 4A

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T?

The displacement is Dx = x2–x1. Since the initial and final positions of the mass are the same (it ends up back at its original position), then the displacement is zero.

Follow-up: What is the net displacement after a half of a period?

ConcepTest 11.1cHarmonic Motion III

A mass on a spring in SHM has amplitude A and period T. How long does it take for the mass to travel a total distance of 6A?

1) 1/2 T

2) 3/4 T

3) 1 1/4 T

4) 1 1/2 T

5) 2 T

ConcepTest 11.1cHarmonic Motion III

A mass on a spring in SHM has amplitude A and period T. How long does it take for the mass to travel a total distance of 6A?

1) 1/2 T

2) 3/4 T

3) 1 1/4 T

4) 1 1/2 T

5) 2 T

We have already seen that it takes one period T to travel a total distance of 4A. An additional 2A requires half a period, so the total time needed for a total distance of 6A is 1 1/2 T.

Follow-up: What is the net displacement at this particular time?

Vibrating Mass on a Spring

- The radius of a circle is symbolic of the amplitude of a wave.
- Energy is conserved as the elastic potential energy in a spring can be converted into kinetic energy. Once again the displacement of a spring is symbolic of the amplitude of a wave
- Since BOTH algebraic expressions have the ratio of the Amplitude to the velocity we can set them equal to each other.
- This derives the PERIOD of a SPRING.

Simple Pendulum

- Derivation is a bit complicated, but I’ll include it here:

Pendulums

Consider the FBD for a pendulum. Here we have the weight and tension. Even though the weight isn’t at an angle let’s draw an axis along the tension.

q

mgcosq

q

mgsinq

Pendulums

What is x? It is the amplitude! In the picture to the left, it represents the chord from where it was released to the bottom of the swing (equilibrium position).

Practice Problem

- A visitor to a lighthouse wishes to determine the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.40 s. What is the height of the tower?

ConcepTest 11.5aEnergy in SHM I

A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

1) total energy will increase

2) total energy will not change

3) total energy will decrease

ConcepTest 11.5aEnergy in SHM I

A mass oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

1) total energy will increase

2) total energy will not change

3) total energy will decrease

The total energy is equal to the initial value of the elastic potential energy, which is PEs = 1/2 kA2. This does not depend on mass, so a change in mass will not affect the energy of the system.

Follow-up: What happens if you double the amplitude?

ConcepTest 11.6aPeriod of a Spring I

A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?

1) period will increase

2) period will not change

3) period will decrease

ConcepTest 11.6aPeriod of a Spring I

A glider with a spring attached to each end oscillates with a certain period. If the mass of the glider is doubled, what will happen to the period?

1) period will increase

2) period will not change

3) period will decrease

The period is proportional to the square root of the mass. So an increase in mass will lead to an increase in the period of motion.

Follow-up: What happens if the amplitude is doubled?

ConcepTest 11.7aSpring in an Elevator I

A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is moving upward at constant speed?

1) period will increase

2) period will not change

3) period will decrease

ConcepTest 11.7aSpring in an Elevator I

A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is moving upward at constant speed?

1) period will increase

2) period will not change

3) period will decrease

Nothing at all changes when the elevator moves at constant speed. The equilibrium elongation of the spring is the same, and the period of simple harmonic motion is the same.

ConcepTest 11.7bSpring in an Elevator II

A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward?

1) period will increase

2) period will not change

3) period will decrease

ConcepTest 11.7bSpring in an Elevator II

A mass is suspended from the ceiling of an elevator by a spring. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward?

1) period will increase

2) period will not change

3) period will decrease

When the elevator accelerates upward, the hanging mass feels “heavier” and the spring will stretch a bit more. Thus, the equilibrium elongation of the spring will increase. However, the period of simple harmonic motion does not depend upon the elongation of the spring – it only depends on the mass and the spring constant, and neither one of them has changed.

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