Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates due to a restoring force that is proportional to its displacement. This force always acts toward the equilibrium position, making SHM fundamental in pendulums, mass-spring systems, and wave motion—key topics in AP® Physics 1. One of the most effective ways to analyze SHM is through graphs of displacement, velocity, and acceleration. These visual representations help students understand how position, speed, and force change over time, reinforcing key physics concepts. By interpreting SHM graphs, students can develop a deeper intuition for how oscillatory motion behaves.

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion is a type of periodic motion where an object moves back and forth over the same path. Importantly, it takes the same amount of time for each complete cycle of movement.

Key Characteristics of SHM

• Oscillatory Motion: SHM involves repetitive back and forth movements.
• Restoring Force: The motion is driven by a force that always points towards the equilibrium position.
• Hooke’s Law: Often used to describe SHM in springs: F = -kx.

Real-World Examples of SHM

Examples of Simple Harmonic Motion (SHM) include:

• A child swinging on a playground swing – The restoring force is gravity, which pulls the child back toward the lowest point of the swing’s arc.
• The pendulum of a clock – The restoring force is the gravitational force acting on the pendulum bob, directed toward the equilibrium position.
• The oscillations of a tuning fork – The restoring force comes from the elastic forces within the metal, which pull the prongs back to their original position after being displaced.

In each case, the restoring force is proportional to displacement and acts in the opposite direction, ensuring periodic motion characteristic of SHM.

Basic Equations of SHM

Two standard equations describe the displacement of SHM at any time (t):

1. x = A \cos(2\pi ft)
2. x = A \sin(2\pi ft)

Definitions of Terms

• Amplitude (A): The maximum displacement from the equilibrium position.
• Frequency (f): The number of oscillations per second, measured in Hertz (Hz).
• Period (T): The time it takes to complete one cycle, calculated as T = \frac{1}{f}.

Types of SHM Graphs

Displacement vs. Time Graph

This graph shows how the position of an object changes over time. It is sinusoidal, reflecting SHM’s oscillatory nature. Key features include:

• Peaks at maximum and minimum displacements (amplitude).
• Zeros when the object passes through the equilibrium position.

Example: Creating a Displacement Graph

Imagine a pendulum with an amplitude of 10 cm and a frequency of 1 Hz. Let’s create its graph over a cycle:

1. Use x = A \sin(2\pi ft).
2. Calculate (x) at different times: t = 0, 0.25, 0.5, 0.75, 1.
3. x(0) = 10\sin(0) = 0, x(0.25) = 10\sin(\frac{\pi}{2}) = 10, x(0.5) = 10\sin(\pi) = 0, and so forth.
4. Plot these points; connect with a smooth sinusoidal curve.

Velocity vs. Time Graph

In SHM, velocity shows the rate of change of displacement. A velocity graph can be derived by differentiating the displacement function. It’s typically a sine or cosine function, too. Notice that:

• Velocity is zero at maximum displacement points.
• It is maximum when passing through equilibrium.

Impact of Amplitude on SHM

Amplitude affects the height of the graph but not the period or frequency. Larger amplitude means greater displacement but the cycle time remains constant.

Example: Comparing Two SHM Graphs

Consider two pendulums with amplitudes of 5 cm and 10 cm, but both have the frequency of 1 Hz. Their graphs have different peak heights, but the same curve repeats every second.

Conclusion: SHM Graphs of Motion

Interpreting SHM graphs is essential for understanding oscillatory motion, wave behavior, and energy transformations in AP® Physics 1. By analyzing displacement, velocity, and acceleration over time, students can develop a deeper intuition for how objects move in Simple Harmonic Motion.

Tips for Studying SHM Graphs:

• Identify Key Patterns – Recognize how displacement, velocity, and acceleration graphs relate to one another.
• Use Simulations – Interactive SHM simulators allow you to manipulate variables like mass, amplitude, and spring constant to see real-time graph changes.
• Practice Problem-Solving – Sketch SHM graphs for different scenarios and predict how motion will change with varying conditions.
• Connect Graphs to Equations – Use formulas like x = A\cos(\omega t) to relate graphical behavior to mathematical expressions.

By combining conceptual understanding with hands-on simulations and practice, students can reinforce their problem-solving skills and prepare effectively for AP® Physics 1 questions on oscillatory motion.

Sharpen Your Skills for AP® Physics 1

Are you preparing for the AP® Physics 1 test? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world physics problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

• AP® Physics 1: 7.1 Review
• AP® Physics 1: 7.2 Review

Need help preparing for your AP® Physics 1 exam?

Albert has hundreds of AP® Physics 1 practice questions, free response, and full-length practice tests to try out.