AP Physics C: Mechanics

Oscillations – The Sinusoidal Description of Simple Harmonic Motion

1. Introduction

Simple Harmonic Motion (SHM) is a type of periodic motion in which an object oscillates back and forth about an equilibrium position.

One of the most important features of SHM is that the motion can be described mathematically using sine and cosine functions.

Because the position, velocity, and acceleration all vary sinusoidally with time, SHM is often called sinusoidal motion.

Examples include:

• A mass attached to a spring

• A simple pendulum at small angles

• Vibrating molecules

• Oscillating mechanical systems

2. Why SHM Is Sinusoidal

The defining equation of SHM is:

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

This equation states that acceleration is proportional to displacement and opposite in direction.

The mathematical functions whose second derivatives are proportional to themselves are sine and cosine functions.

Therefore, SHM is naturally described by sinusoidal functions.

3. Position as a Function of Time

The displacement of an object undergoing SHM is:

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

where:

• (A) = amplitude

• (\omega) = angular frequency

• (t) = time

• (\phi) = phase constant

This equation completely describes the object’s position at any moment.

4. Meaning of Amplitude

Amplitude

The amplitude is the maximum displacement from equilibrium.

Symbol:

$$
A
$$

Units:

$$
m
$$

The object oscillates between:

$$
+A
$$

and

$$
-A
$$

5. Meaning of Angular Frequency

Angular Frequency

Angular frequency determines how rapidly the oscillation occurs.

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

or

$$
\omega=2\pi f
$$

where:

• (T) = period

• (f) = frequency

Larger values of (\omega) correspond to faster oscillations.

6. Meaning of the Phase Constant

Phase Constant

The phase constant determines the starting position of the oscillator.

Symbol:

$$
\phi
$$

Different values of (\phi) correspond to different initial conditions.

Example

If

$$
\phi=0
$$

then

$$
x(0)=A
$$

The oscillator begins at maximum displacement.

7. Alternative Form of the Position Equation

The position can also be written as:

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

Both sine and cosine forms describe SHM.

The choice depends on the initial conditions.

8. Velocity as a Function of Time

Velocity is the derivative of position.

Starting with:

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

Differentiate with respect to time:

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

Maximum Velocity

The maximum speed is:

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

This occurs when the object passes through equilibrium.

9. Acceleration as a Function of Time

Acceleration is the derivative of velocity.

Starting with:

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

Differentiate again:

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

Since

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

we obtain:

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

10. Maximum Acceleration

The largest acceleration occurs at maximum displacement.

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

This occurs at:

$$
x=\pm A
$$

11. Phase Relationships

The position, velocity, and acceleration do not reach their maximum values at the same time.

Position and Velocity

Position:

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

Velocity:

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

Velocity is shifted by:

$$
90^\circ
$$

or

$$
\frac{\pi}{2}
\text{ radians}
$$

relative to position.

Position and Acceleration

Acceleration:

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

Acceleration is exactly opposite in phase to displacement.

They differ by:

$$
180^\circ
$$

or

$$
\pi
\text{ radians}
$$

12. Graph of Position vs. Time

The position graph is a cosine wave.

Characteristics:

• Smooth and continuous

• Repeats every period

• Maximum value is (+A)

• Minimum value is (-A)

The graph repeats every:

$$
T
$$

seconds.

13. Graph of Velocity vs. Time

The velocity graph is also sinusoidal.

Characteristics:

• Same period as position

• Shifted by one-quarter cycle

• Maximum at equilibrium

The velocity graph leads or lags the position graph by:

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

14. Graph of Acceleration vs. Time

The acceleration graph is sinusoidal.

Characteristics:

• Same period

• Opposite phase from displacement

• Maximum when displacement is maximum

Because:

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

the acceleration graph is an inverted version of the position graph.

15. Position, Velocity, and Acceleration at Key Locations

At Maximum Displacement

$$
x=\pm A
$$

$$
v=0
$$

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

At Equilibrium

$$
x=0
$$

$$
v=\pm A\omega
$$

$$
a=0
$$

16. Velocity–Position Relationship

Eliminating time from the equations gives:

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

This equation is extremely useful in AP Physics C problems.

17. Example Problem

A spring oscillator has:

$$
A=0.15,m
$$

and

$$
\omega=8,rad/s
$$

Find the maximum velocity.

Solution

Using:

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

Substitute values:

$$
v_{\max}=(0.15)(8)
$$

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

18. Example Problem

Using the same oscillator, find the maximum acceleration.

Solution

Using:

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

Substitute values:

$$
a_{\max}=(0.15)(8^2)
$$

$$
a_{\max}=(0.15)(64)
$$

$$
a_{\max}=9.6,m/s^2
$$

19. Common AP Physics C Mistakes

Mistake 1

Forgetting that velocity and position are out of phase.

When position is maximum:

$$
v=0
$$

Mistake 2

Assuming acceleration is maximum at equilibrium.

Actually:

$$
a=0
$$

at equilibrium.

Mistake 3

Using incorrect units for angular frequency.

Angular frequency is measured in:

$$
rad/s
$$

not hertz.

20. AP Physics C Exam Tips

Memorize These Equations

Position:

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

Velocity:

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

Acceleration:

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

Angular Frequency:

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

Maximum Velocity:

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

Maximum Acceleration:

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

Summary

The sinusoidal description of SHM uses sine and cosine functions to model oscillatory motion.

Key equations:

Position:

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

Velocity:

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

Acceleration:

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

Velocity-Position Relation:

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

Key ideas:

• SHM is sinusoidal motion.

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

• Velocity is 90° out of phase with position.

• Acceleration is 180° out of phase with position.

• Sinusoidal equations provide a complete mathematical description of oscillatory motion.