Oscillations – The Kinematics of Simple Harmonic Motion

1. Introduction

The kinematics of Simple Harmonic Motion (SHM) describes how an oscillating object’s position, velocity, and acceleration change with time.

Unlike constant-acceleration motion, SHM involves continuously changing velocity and acceleration.

Examples of SHM include:

• a mass attached to a spring
• a pendulum at small angles
• vibrating molecules
• oscillating mechanical systems

The motion is periodic and repeats itself after a fixed interval of time.

2. What Is Simple Harmonic Motion?

Definition

Simple Harmonic Motion is motion in which the acceleration is directly proportional to displacement and opposite in direction.

a=−ω2xa=-\omega^2xa=−ω2x

where:

• $a$ = acceleration
• $x$ = displacement from equilibrium
• $\omega$ = angular frequency

The negative sign indicates that the acceleration always points toward the equilibrium position.

3. The Oscillation Cycle

During one complete cycle:

1. The object moves from one turning point.
2. Passes through equilibrium.
3. Reaches the opposite turning point.
4. Returns to its starting position.

This repeating motion defines an oscillation.

4. Amplitude

Definition

Amplitude is the maximum displacement from equilibrium.

Symbol:

$$A$$

Units:

$$m$$

The object oscillates between:

$$+A$$

and

$$-A$$

5. Period

Definition

The period is the time required for one complete oscillation.

Symbol:

$$T$$

Units:

$$s$$

After one period:

$$x(t+T)=x(t)$$

The motion repeats exactly.

6. Frequency

Definition

Frequency is the number of oscillations per second.

f=1Tf = \frac{1}{T}f=T1​

Units:

$$Hz$$

7. Angular Frequency

Angular frequency measures how rapidly the oscillation occurs.

$$\omega =2\pi f$$

or

ω=2πT\omega = \frac{2\pi}{T}ω=T2π​

Interpretation

Larger values of $\omega$ correspond to faster oscillations.

8. Position as a Function of Time

The displacement in SHM is described by a sinusoidal function.

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

where:

• $A$ = amplitude
• $\omega$ = angular frequency
• $\varphi$ = phase constant

Alternative Form

The motion can also be written as:

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

Both equations describe SHM.

9. Position Graph

The position-versus-time graph is a sine or cosine curve.

Characteristics:

• periodic
• smooth
• symmetric

The graph repeats every period $T$.

10. Velocity in SHM

Velocity is the derivative of position.

Starting with:

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

Differentiating:

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

11. Maximum Velocity

The maximum speed occurs at equilibrium.

vmax⁡=Aωv_{\max} = A\omegavmax​=Aω

At Equilibrium

$$x=0$$

Velocity reaches its greatest magnitude.

12. Velocity Graph

The velocity graph is also sinusoidal.

It is shifted by one-quarter cycle relative to the position graph.

Position and velocity are always out of phase by:

90∘90^\circ90∘

or

π2\frac{\pi}{2}2π​

radians.

13. Acceleration in SHM

Acceleration is the derivative of velocity.

Starting with:

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

Differentiating:

a(t)=−Aω2cos⁡(ωt+ϕ)a(t) = -A\omega^2 \cos(\omega t+\phi)a(t)=−Aω2cos(ωt+ϕ)

Since:

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

we obtain:

a=−ω2xa=-\omega^2xa=−ω2x

14. Maximum Acceleration

Acceleration is greatest at the turning points.

amax⁡=ω2Aa_{\max} = \omega^2Aamax​=ω2A

At Turning Points

$$x=\pm A$$

Velocity is zero.

Acceleration is maximum.

15. Phase Relationships

The three kinematic variables are not maximized at the same time.

Position

Maximum at:

$$x=\pm A$$

Velocity

Maximum at:

$$x=0$$

Acceleration

Maximum at:

$$x=\pm A$$

16. Summary Table

17. Velocity–Position Relationship

A useful equation relates velocity and position directly.

v2=ω2(A2−x2)v^2 = \omega^2(A^2-x^2)v2=ω2(A2−x2)

This equation eliminates time from the analysis.

Special Cases

At equilibrium:

$$x=0$$ v=vmax⁡=Aωv=v_{\max}=A\omegav=vmax​=Aω

At a turning point:

$$x=A$$ $$v=0$$

18. Example Problem

A block oscillates with:

$$A=0.20m$$

and

$$\omega =5rad/s$$

Find the maximum speed.

Solution

Using:

vmax⁡=Aωv_{\max} = A\omegavmax​=Aω

Substitute values:

vmax⁡=(0.20)(5)v_{\max} = (0.20)(5)vmax​=(0.20)(5) vmax⁡=1.0 m/sv_{\max} = 1.0\,m/svmax​=1.0m/s

19. Example Problem

Using the previous oscillator, find the maximum acceleration.

Solution

Using:

amax⁡=ω2Aa_{\max} = \omega^2Aamax​=ω2A

Substitute values:

amax⁡=(5)2(0.20)a_{\max} = (5)^2(0.20)amax​=(5)2(0.20) amax⁡=25(0.20)a_{\max} = 25(0.20)amax​=25(0.20) amax⁡=5.0 m/s2a_{\max} = 5.0\,m/s^2amax​=5.0m/s2

20. Graphical Interpretation

Position Graph

Cosine or sine wave.

Velocity Graph

Shifted by one-quarter period.

Acceleration Graph

Opposite in sign to position.

Because:

a=−ω2xa=-\omega^2xa=−ω2x

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

21. Common AP Physics C Mistakes

Mistake 1

Assuming acceleration is greatest at equilibrium.

Actually:

$$a=0$$

at equilibrium.

Mistake 2

Assuming velocity is greatest at maximum displacement.

Velocity is zero at the turning points.

Mistake 3

Forgetting the negative sign in:

a=−ω2xa=-\omega^2xa=−ω2x

The negative sign indicates the restoring nature of SHM.

22. AP Physics C Connections

Calculus

Velocity is the derivative of position.

Acceleration is the derivative of velocity.

Dynamics

SHM results from restoring forces.

Energy

The kinematic variables are directly related to kinetic and potential energy.

Oscillations

The same kinematic principles apply to springs, pendulums, and many other oscillating systems.

Summary

Position:

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

Velocity:

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

Acceleration:

a(t)=−ω2xa(t) = -\omega^2xa(t)=−ω2x

Angular frequency:

ω=2πT\omega = \frac{2\pi}{T}ω=T2π​

Maximum velocity:

vmax⁡=Aωv_{\max} = A\omegavmax​=Aω

Maximum acceleration:

amax⁡=ω2Aa_{\max} = \omega^2Aamax​=ω2A

Velocity-position relation:

v2=ω2(A2−x2)v^2 = \omega^2(A^2-x^2)v2=ω2(A2−x2)

Key ideas:

• SHM is sinusoidal motion
• position, velocity, and acceleration vary continuously with time
• velocity is maximum at equilibrium
• acceleration is maximum at the turning points
• acceleration is always directed toward equilibrium

The kinematics of Simple Harmonic Motion provides the mathematical description of oscillations and serves as the foundation for analyzing springs, pendulums, waves, and many advanced topics in AP Physics C Mechanics.