AP Physics C: Mechanics

Oscillations – Motion in Simple Harmonic Motion

1. Introduction

An oscillation is a repetitive back-and-forth motion about an equilibrium position.

In AP Physics C, the most important type of oscillatory motion is Simple Harmonic Motion (SHM).

The motion of an object in SHM can be analyzed by studying:

• Position

• Velocity

• Acceleration

• Force

• Energy

All of these quantities change continuously as the object oscillates.

2. Equilibrium Position

Definition

The equilibrium position is the location where the net force acting on the object is zero.

$$
\sum F = 0
$$

At equilibrium:

$$
a = 0
$$

An object in SHM repeatedly moves through this position.

3. Displacement

Definition

Displacement is the distance and direction from equilibrium.

Symbol:

$$
x
$$

Units:

$$
m
$$

The displacement changes continuously during the motion.

4. Amplitude

Definition

Amplitude is the maximum displacement from equilibrium.

Symbol:

$$
A
$$

The oscillator moves between:

$$
+A
$$

and

$$
-A
$$

Important Note

Amplitude is always positive.

It represents the greatest distance from equilibrium.

5. Period

Definition

The period is the time required for one complete oscillation.

Symbol:

$$
T
$$

Units:

$$
s
$$

The motion repeats every period.

6. Frequency

Definition

Frequency is the number of oscillations completed per second.

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

Units:

$$
Hz
$$

7. Angular Frequency

Angular frequency describes how rapidly the oscillation occurs.

$$
\omega = 2\pi f
$$

or

$$
\omega = \frac{2\pi}{T}
$$

Units:

$$
rad/s
$$

8. Position During SHM

The displacement as a function of time is:

$$
x(t)=A\cos(\omega t+\phi)
$$

where:

• (A) = amplitude

• (\omega) = angular frequency

• (\phi) = phase constant

This equation describes the object’s motion at all times.

9. Motion at the Turning Points

The turning points occur at:

$$
x=+A
$$

and

$$
x=-A
$$

Characteristics

At the turning points:

$$
v=0
$$

Velocity is zero because the object changes direction.

Acceleration is maximum.

Force is maximum.

10. Motion at Equilibrium

At equilibrium:

$$
x=0
$$

Characteristics

Velocity reaches its greatest magnitude.

$$
v=v_{max}
$$

Acceleration becomes:

$$
a=0
$$

The restoring force is also zero.

11. Velocity During SHM

Velocity is the derivative of position.

Starting with:

$$
x(t)=A\cos(\omega t+\phi)
$$

Differentiating:

$$
v(t)=-A\omega\sin(\omega t+\phi)
$$

12. Maximum Velocity

The maximum speed occurs when the object passes through equilibrium.

$$
v_{max}=A\omega
$$

Important Observation

The larger the amplitude, the greater the maximum speed.

The larger the angular frequency, the greater the maximum speed.

13. Acceleration During SHM

Acceleration is the derivative of velocity.

$$
a(t)=-A\omega^2\cos(\omega t+\phi)
$$

Since

$$
x(t)=A\cos(\omega t+\phi)
$$

we obtain:

$$
a=-\omega^2x
$$

14. Meaning of the Acceleration Equation

The equation

$$
a=-\omega^2x
$$

shows that:

• Acceleration is proportional to displacement.

• Acceleration points toward equilibrium.

• The farther the object is from equilibrium, the larger the acceleration.

15. Maximum Acceleration

The largest acceleration occurs at the turning points.

$$
a_{max}=A\omega^2
$$

This occurs when:

$$
x=\pm A
$$

16. Restoring Force

The force responsible for SHM is called the restoring force.

For a spring:

$$
F=-kx
$$

The negative sign indicates that the force always acts toward equilibrium.

17. Relationship Between Position, Velocity, and Acceleration

The motion follows a predictable pattern.

At Maximum Displacement

$$
x=\pm A
$$

$$
v=0
$$

$$
a=\mp A\omega^2
$$

At Equilibrium

$$
x=0
$$

$$
v=\pm A\omega
$$

$$
a=0
$$

18. Velocity–Position Relationship

Removing time from the equations gives:

$$
v^2=\omega^2(A^2-x^2)
$$

This equation is useful when position is known but time is not.

19. Motion Through One Complete Cycle

Consider an oscillator released from:

$$
x=A
$$

Quarter Cycle

After:

$$
\frac{T}{4}
$$

the object reaches equilibrium.

Velocity is maximum.

Half Cycle

After:

$$
\frac{T}{2}
$$

the object reaches:

$$
x=-A
$$

Velocity is zero.

Three-Quarter Cycle

After:

$$
\frac{3T}{4}
$$

the object passes equilibrium again.

Velocity is maximum.

Full Cycle

After:

$$
T
$$

the object returns to its original position.

20. Example Problem

A mass oscillates with:

$$
A=0.20,m
$$

and

$$
\omega=6,rad/s
$$

Find the maximum velocity.

Solution

Using:

$$
v_{max}=A\omega
$$

Substitute values:

$$
v_{max}=(0.20)(6)
$$

$$
v_{max}=1.2,m/s
$$

21. Example Problem

Find the maximum acceleration.

Solution

Using:

$$
a_{max}=A\omega^2
$$

Substitute values:

$$
a_{max}=(0.20)(6^2)
$$

$$
a_{max}=(0.20)(36)
$$

$$
a_{max}=7.2,m/s^2
$$

22. Common AP Physics C Mistakes

Mistake 1

Assuming velocity is greatest at maximum displacement.

Actually:

$$
v=0
$$

at the turning points.

Mistake 2

Assuming acceleration is greatest at equilibrium.

Actually:

$$
a=0
$$

at equilibrium.

Mistake 3

Forgetting the negative sign in:

$$
a=-\omega^2x
$$

The negative sign represents the restoring nature of SHM.

23. AP Physics C Exam Tips

Memorize These Relationships

Position:

$$
x(t)=A\cos(\omega t+\phi)
$$

Velocity:

$$
v(t)=-A\omega\sin(\omega t+\phi)
$$

Acceleration:

$$
a(t)=-A\omega^2\cos(\omega t+\phi)
$$

Maximum Velocity:

$$
v_{max}=A\omega
$$

Maximum Acceleration:

$$
a_{max}=A\omega^2
$$

Velocity–Position Relation:

$$
v^2=\omega^2(A^2-x^2)
$$

Summary

Motion in Simple Harmonic Motion is characterized by periodic oscillations about an equilibrium position.

Key equations:

Position:

$$
x(t)=A\cos(\omega t+\phi)
$$

Velocity:

$$
v(t)=-A\omega\sin(\omega t+\phi)
$$

Acceleration:

$$
a=-\omega^2x
$$

Maximum Velocity:

$$
v_{max}=A\omega
$$

Maximum Acceleration:

$$
a_{max}=A\omega^2
$$

Key ideas:

• The object oscillates between two turning points.

• Velocity is maximum at equilibrium.

• Acceleration is maximum at the turning points.

• The restoring force always points toward equilibrium.

• Position, velocity, and acceleration vary sinusoidally with time.

Understanding the motion of SHM is essential for analyzing springs, pendulums, waves, and many advanced applications of oscillatory systems in AP Physics C Mechanics.