1. Introduction

A spring–block oscillator is one of the most important examples of Simple Harmonic Motion (SHM).

In this system, a block attached to a spring moves back and forth about an equilibrium position.

The motion repeats at regular intervals and is caused by a restoring force that always points toward the equilibrium position.

Examples include:

• a mass attached to a spring
• vibration of molecules
• vehicle suspension systems
• mechanical oscillators

2. What Is Simple Harmonic Motion?

Definition

Simple Harmonic Motion (SHM) is periodic motion in which the restoring force is directly proportional to displacement and opposite in direction.

Mathematically:

$$F=-kx$$

where:

• $F$ = restoring force
• $k$ = spring constant
• $x$ = displacement from equilibrium

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

3. Hooke’s Law

The force exerted by an ideal spring is given by Hooke’s Law.

where:

• $k$ = spring constant $(N/m)$
• $x$ = displacement from equilibrium

A larger spring constant means a stiffer spring.

4. Equilibrium Position

The equilibrium position is the location where:

$$F=0$$

At equilibrium:

$$x=0$$

If the block is displaced from this position, the spring exerts a restoring force that pulls it back.

5. Newton’s Second Law and SHM

Applying Newton’s Second Law:

$$\sum F=ma$$

Using Hooke’s Law:

$$-kx=ma$$

Therefore:

a=−kmxa = -\frac{k}{m}xa=−mk​x

Key Observation

Acceleration is proportional to displacement and opposite in direction.

This relationship defines simple harmonic motion.

6. Characteristics of SHM

The motion is:

• periodic
• oscillatory
• symmetric about equilibrium

The block repeatedly moves between two turning points.

7. Amplitude

Definition

Amplitude is the maximum displacement from equilibrium.

Symbol:

$$A$$

Units:

$$m$$

The block oscillates between:

$$+A$$

and

$$-A$$

8. Period

Definition

The period is the time required for one complete oscillation.

Symbol:

$$T$$

Units:

$$s$$

For a spring–block oscillator:

T=2πmkT = 2\pi \sqrt{\frac{m}{k}}T=2πkm​​

Observations

The period increases when:

• mass increases

The period decreases when:

• spring stiffness increases

9. Frequency

Definition

Frequency is the number of oscillations per second.

f=1Tf = \frac{1}{T}f=T1​

Units:

$$Hz$$

For a spring system:

f=12πkmf = \frac{1}{2\pi} \sqrt{\frac{k}{m}}f=2π1​mk​​

10. Angular Frequency

Angular frequency is:

$$\omega =2\pi f$$

For a spring oscillator:

ω=km\omega = \sqrt{\frac{k}{m}}ω=mk​​

Important Relationship

This equation is one of the most frequently tested SHM formulas.

11. Position as a Function of Time

The displacement can be described by:

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

where:

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

This equation completely describes the motion.

12. Velocity in SHM

Velocity is the derivative of position.

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

Maximum Velocity

The greatest speed occurs at equilibrium.

vmax⁡=Aωv_{\max} = A\omegavmax​=Aω

13. Acceleration in SHM

Acceleration is:

a(t)=−ω2xa(t) = -\omega^2xa(t)=−ω2x

Maximum Acceleration

At the turning points:

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

Acceleration is largest when displacement is largest.

14. Position, Velocity, and Acceleration

At Equilibrium

$$x=0$$

Velocity is maximum.

v=vmax⁡v=v_{\max}v=vmax​

Acceleration is zero.

$$a=0$$

At Turning Points

$$x=\pm A$$

Velocity is zero.

$$v=0$$

Acceleration is maximum.

a=±amax⁡a=\pm a_{\max}a=±amax​

15. Energy in SHM

Mechanical energy remains constant.

$$E=K+U$$

where:

• $K$ = kinetic energy
• $U$ = spring potential energy

16. Spring Potential Energy

The energy stored in a spring is:

U=12kx2U = \frac12 kx^2U=21​kx2

At maximum displacement:

Umax⁡=12kA2U_{\max} = \frac12 kA^2Umax​=21​kA2

17. Kinetic Energy

The kinetic energy is:

K=12mv2K = \frac12 mv^2K=21​mv2

At equilibrium:

Kmax⁡=12kA2K_{\max} = \frac12 kA^2Kmax​=21​kA2

18. Total Mechanical Energy

The total energy remains constant.

E=12kA2E = \frac12 kA^2E=21​kA2

Important Result

Total energy depends on:

• spring constant
• amplitude

Total energy does not depend on time.

19. Energy Transformation

As the block moves:

• potential energy converts into kinetic energy
• kinetic energy converts into potential energy

The total energy remains constant.

At Maximum Displacement

$$K=0$$ $$U=E$$

At Equilibrium

$$U=0$$ $$K=E$$

20. Example Problem

A block of mass:

$$m=2.0kg$$

is attached to a spring with:

$$k=200N/m$$

Find the period.

Solution

Using:

T=2πmkT = 2\pi \sqrt{\frac{m}{k}}T=2πkm​​

Substitute values:

T=2π2.0200T = 2\pi \sqrt{\frac{2.0}{200}}T=2π2002.0​​ T=2π0.01T = 2\pi \sqrt{0.01}T=2π0.01​ $$T=2\pi (0.1)$$ $$T\approx 0.628s$$

21. Common AP Physics C Mistakes

Mistake 1

Using equilibrium as the point of maximum acceleration.

At equilibrium:

$$a=0$$

Mistake 2

Assuming velocity is greatest at maximum displacement.

Velocity is zero at the turning points.

Mistake 3

Confusing amplitude with total distance traveled.

Amplitude is measured from equilibrium to a turning point.

22. AP Physics C Connections

Newton’s Second Law

$$F=ma$$

combined with Hooke’s Law produces SHM.

Energy Conservation

Explains energy exchange during oscillation.

Calculus

Velocity and acceleration are derivatives of position.

Oscillations

The spring–block system serves as the model for many oscillatory systems.

Summary

Hooke’s Law:

$$F=-kx$$

Angular frequency:

ω=km\omega = \sqrt{\frac{k}{m}}ω=mk​​

Period:

T=2πmkT = 2\pi \sqrt{\frac{m}{k}}T=2πkm​​

Frequency:

f=12πkmf = \frac{1}{2\pi} \sqrt{\frac{k}{m}}f=2π1​mk​​

Spring potential energy:

U=12kx2U = \frac12 kx^2U=21​kx2

Total energy:

E=12kA2E = \frac12 kA^2E=21​kA2

Key ideas:

• SHM occurs when the restoring force is proportional to displacement
• acceleration always points toward equilibrium
• velocity is maximum at equilibrium
• acceleration is maximum at the turning points
• total mechanical energy remains constant

The spring–block oscillator is the fundamental model of Simple Harmonic Motion and serves as the basis for understanding oscillations throughout AP Physics C Mechanics.