Simple Harmonic Motion: Kinetic & Potential Energy Exchange

Interactive Simple Harmonic Motion (SHM) Simulation Guide

This simulation visualizes the energy transformations and kinematics of a mass-spring system oscillating on a frictionless horizontal surface. Use the controls to modify system parameters and observe how they affect the period and energy distribution.

1. Controls & Interface

• Start / Stop: Toggles the simulation.
• Step →: Advances the simulation by small time increments (Δt = 0.1 s). Use this to freeze the motion at specific points (like maximum extension) to analyze the energy values.
• System Parameters:
  • Amplitude (A): The maximum displacement from the equilibrium position.
  • Mass (m): The inertia of the oscillating block.
  • Spring Constant (k): The stiffness of the spring. Higher values mean a stiffer spring.
  • Duration: Sets how long the graph records data before resetting or stopping.
• Graph Toggles: Use the checkboxes below the graph to hide or show Kinetic Energy (KE), Potential Energy (PE), or Total Energy (TE) curves individually.

2. Key Insights to Observe

A. Conservation of Energy

Watch the Total Energy (TE) line (green). It remains perfectly horizontal and constant throughout the motion. This confirms that in an ideal SHM system without friction or damping, energy is never lost; it simply transforms form.

• TE = ½kA² = constant

B. Energy Exchange

Observe the Kinetic (Orange) and Potential (Purple) energy bars on the right.

• At Equilibrium (x = 0): The spring is neither stretched nor compressed. Potential Energy is zero. Kinetic Energy is at its maximum. The block is moving fastest here.
• At Extremes (x = ±A): The block momentarily stops to reverse direction. Kinetic Energy drops to zero. Potential Energy is at its maximum because the spring deformation is greatest.

C. Factors Affecting Time Period (T)

Experiment by changing Mass (m) and Spring Constant (k) separately.

• Increase Mass: You will see the oscillation slow down. The Period (T) increases because higher inertia resists acceleration.
• Increase Spring Constant: You will see the oscillation speed up. The Period (T) decreases because a stiffer spring provides a larger restoring force.
• Change Amplitude: Change A from 10 to 50. Notice that the Period (T) does not change. This demonstrates the isochronous nature of SHM—the time period is independent of amplitude.

D. Frequency of Energy Oscillation

Look at the graph. In one complete oscillation of the block (one full period T), the Kinetic Energy curve hits its peak twice (once moving right through the center, once moving left).

• Insight: The frequency of the energy oscillation is 2× the frequency of the position oscillation.

3. Mathematical Relationships

Verify the simulation data against these standard equations:

• Period: T = 2π√(m/k)
• Total Energy: E = ½kA²
• Potential Energy: U = ½kx²
• Kinetic Energy: K = ½mv²

How do kinetic and potential energy trade off in simple harmonic motion?

In short: In a frictionless mass–spring system, the total energy E = ½kA² stays constant while it shifts between spring potential energy U = ½kx² and kinetic energy K = ½mv². Potential energy is largest at the turning points x = ±A, where the mass momentarily stops, and zero at equilibrium. Kinetic energy is the reverse. At every position K + U = ½kA².

Common doubts

• “Is energy lost when the mass stops at the turning point?” No. At that instant all the energy is stored in the spring as potential energy. Nothing is lost, and it turns back into kinetic energy as the mass returns.
• “How often do K and U swap?” Both oscillate twice per period, as mirror images of each other, while the displacement completes one cycle.
• “Does a heavier mass change the total energy?” Not for a given amplitude, since E = ½kA² has no m in it. The mass changes the period, T = 2π√(m ÷ k), and so how quickly the energy swaps.

Change the amplitude, spring constant and mass in the simulation above, press START, and watch the PE, KE and total energy curves.

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