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# Simple Harmonic Motion and Uniform Circular Motion

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## Period and Frequency in Oscillations

When a guitar string is plucked or a spring moves up and down, the time interval between each oscillation is defined as periodic motion. The time that one oscillation is completed is called the period. In addition to indicating repeated oscillations, a period can also represent one event.  The unit of measurement for a period is typically indicated in seconds.

## Frequency

When a period is repeated, the number of oscillations per unit of time is indicated as frequency. Mathematically, the frequency is expressed by the following formula:

$f=\frac{1}{T}$

Where,

f = frequency

T = Time period

1 = Constant

The SI unit for frequency is in cycles per second, also known as hertz (Hz).  One cycle is equal to one oscillation. Oscillations are repetitive for a number of cycles.

Review Questions

1.  A ______ causes a disturbance in a system that activates an oscillation.

2. Waves created by oscillations carry energy. T/F

3. The restoring force of an object ______ when the deformation is increased.

• is increased
• is decreased
• stays the same

4. One cycle is equal to how many oscillations?

5. The term _______ refers to the number of oscillations per unit of time.

## Simple Harmonic Motion: A special Periodic Motion

Oscillations are very common in nature and by human made objects because they occur in so many different ways.  One type of oscillation is simple harmonic motion. It refers to oscillatory motion that is directly proportional to displacement, and the system in which oscillations occur is called a simple harmonic oscillator.

If there is no friction or other nonconservative forces that dampen oscillations, a simple harmonic oscillator will continue to oscillate indefinitely with equal displacement on either side of the equilibrium position.  The maximum displacement from equilibrium is called amplitude.  Amplitude and displacement of objects such as a metal coiled spring are in meters, whereas sound oscillations are indicated in units of pressure.

## Simple Harmonic Motion

A significant fact about simple harmonic motion is that the period t and frequency f are independent of amplitude.  For example, guitar strings will oscillate at the same frequency whether it is plucked gently or hard.  For this reason, simple harmonic oscillations are used to operate clocks because the period remains constant.

The only factors that affect the period and frequency of simple harmonic motion are mass and the force constant k.  Whenever a harmonic oscillator is stiff, a large force k is required to activate it.  Also, it will have a smaller time period than an object that is less stiff.  The period of a harmonic oscillator is impacted by its mass.  The more massive the system is, the longer its period.

## The Link between Harmonic Motion and Waves

All simple harmonic motions are related to sine and cosine waves.  The displacement is a function of time in any harmonic motion as oscillations occur with a period T.  The velocity of the motion is also a function of time.  At maximum displacement from equilibrium, velocity and time are zero.

## The Simple Pendulum

One type of simple harmonic oscillator is a simple pendulum. A simple pendulum is an object that has a small mass, which is suspended by a light wire or string.

When a simple pendulum is displaced from equilibrium, it swings in an arc.  The length of the displacement is called the arc length and is identified as s. When displacement occurs, a restoring force is created that is in the direction towards the equilibrium position.  This restoring force is directly proportional to the displacement.

Two factors affect the period of a simple pendulum, which is the time duration at which one oscillation takes place.  One factor is the length of the string or wire, and the second factor is the acceleration due to gravity.  The period T is nearly independent of amplitude and mass.

Fig.1: Simple Pendulum – Harmonic Oscillator

Review Questions

6. A simple harmonic motion is never capable of oscillating indefinitely. T/F

7. A significant fact about simple harmonic motion is that _____ is independent of amplitude.

• the period
• frequency
• Both a and b

8. Which factor is true about affecting the period and frequency of simple harmonic motion?

• The less stiff an object is, the smaller its time period.
• Whenever a harmonic oscillator is stiff, a large force is required to activate.
• The more massive a system is, the longer the period.
• All of the above

9. When an object oscillates and reaches its maximum displacement, velocity and time are ______.

10. List two factors that affect the time period of a simple pendulum.

Energy and the Simple Harmonic Oscillator

A simple harmonic oscillator has both potential energy and kinetic energy.  When an object is deformed and at the moment it is not moving, it has stored potential energy.

Because a simple harmonic oscillator has no dissipative forces, it has kinetic energy.  Therefore, as an undamped simple harmonic motion takes place, the energy oscillates back and forth between kinetic and potential energy.  An example is the oscillations of a spring.  When it is completely compressed and is not moving, all energy is stored as potential energy.  When the spring decompresses, the elastic potential energy is converted to kinetic energy.  At the equilibrium position, the entire energy of the spring is kinetic.  As it moves passed equilibrium, the energy in the spring is converting back to potential energy.

## Velocity during Oscillations

When a simple harmonic oscillation has reached its maximum displacement position, the velocity is zero.  In this position, all of the energy is in the potential form and there is no kinetic energy.  As the restoring force causes the oscillation to move towards equilibrium, the potential energy decreases, and the kinetic energy increases.  Energy is shared by both, but the total energy does not change.  When the oscillation reaches the equilibrium position, its velocity is at a maximum level.

Maximum velocity depends on three factors:

1. Maximum velocity is directly proportional to amplitude.
2. Maximum velocity is greater for stiffer objects.
3. Maximum velocity is smaller for objects that have larger masses.

## Uniform Circular Motion

When an object moves in a circular path with a constant angular velocity and uniform circular motion, a simple harmonic motion takes place.  The motion is back and forth on the x-axis.  The period T of an oscillator is the time it takes for the object to make one complete revolution.

When viewing a merry-go-round from a distance, any object exhibits simple harmonic motion when it goes back and forth between left and right positions as it turns to create uniform circular motion.

Review Questions

16. Give an example of when the damping of an oscillator is desirable.

17. As the oscillations of harmonic motion slow down due to damping, the net force _____.

• increases
• decreases
• stays the same

18. _______ damping refers to a system that is slow and sluggish.

• Over
• Under
• Critical

19. When driving an object with a frequency equal to its natural frequency, a condition called ______ occurs.

20. Whenever the damping of a harmonic oscillator becomes smaller, the amplitude of the oscillator also becomes smaller. T/F

1. force
2. T
3. a
4. One
5. Frequency
6. F
7. c
8. d
9. zero
10. Length and acceleration due to gravity
11. c
12. zero
13. T
14. It remains constant
15. A
16. The shocks on an automobile
17. b
18. c
19. resonance
20. F

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