Simple Harmonic Motion: How Mass & Spring Constant Set the Period

Interactive Simple Harmonic Motion (SHM) Simulator

Explore the fundamentals of Simple Harmonic Motion (SHM) with this interactive tool! This simulation visualizes a mass oscillating on a horizontal spring and simultaneously plots its key physical properties in real-time.

How to Use the Simulation:

• Run & Pause: Use the Start/Stop button to run or pause the animation.
• Analyze: The Step → button advances the simulation by a small time (0.1s) for careful analysis of the vectors and graph positions.
• Experiment: Before starting, adjust the parameters in the "Controls" section:
  • Amplitude (A): Sets the maximum displacement from the center (equilibrium) position.
  • Start Pos (x₀): Sets the initial position of the mass at time t = 0. This value must be between -A and +A.
  • Period (T): Sets the total time (in seconds) it takes for one complete oscillation.
  • Graph Duration: Sets the total time (in seconds) shown on the x-axis of the graph below.
• Restart: Press Reset Simulation to apply your new parameters and return the time to zero.

What Students Can Understand:

This simulation is designed to build an intuitive understanding of the core concepts of SHM by connecting the physical motion of the block to its abstract graphs:

1. Phase Relationships: Observe the graphs for Displacement (Cyan), Velocity (Red), and Acceleration (Green) all at once. Notice that:

• • When displacement is maximum (at +A or -A), velocity is zero, and acceleration is maximum but in the opposite direction.
  • When displacement is zero (at the center), velocity is at its maximum, and acceleration is zero.

1. Vector Visualization: The arrows above the block show the instantaneous velocity and acceleration.

• • The red (velocity) vector shrinks as the block approaches an end and grows as it moves toward the center.
  • The green (acceleration) vector always points towards the center equilibrium position and is longest when the block is farthest away (at +A or -A).

Play Around and Discover:

• What happens to the maximum velocity (the peak of the red graph) if you keep the amplitude the same but decrease the period?
• Set the Start Pos (x₀) to 0. How do the graphs change compared to starting at x₀ = A? This demonstrates a 90-degree phase shift.
• Pause the simulation exactly where the green acceleration graph crosses the zero line. Where is the block, and what is its velocity vector doing?

How do displacement, velocity and acceleration relate in simple harmonic motion?

In short: For a mass on a spring, the displacement x varies like a sine or cosine curve, and the acceleration is always a = −(k ÷ m)x, proportional to the displacement and pointing toward equilibrium. The velocity is a quarter of a cycle ahead of the displacement. The period T = 2π√(m ÷ k) is set by the system, not by the amplitude.

Common doubts

• “When x is largest, is the velocity largest too?” No. At x = ±A the velocity is zero and the acceleration is at its largest size. At x = 0 the velocity is largest and the acceleration is zero.
• “Does a bigger amplitude make the period longer?” No. Amplitude does not appear in T = 2π√(m ÷ k). Change A and the period stays the same.
• “How do the mass and the spring change the period?” More mass means more inertia and a longer period. A stiffer spring, with a larger k, gives a stronger restoring force and a shorter period.

Change the amplitude, mass and spring constant in the simulation above, press START, and watch x, v and a on the graph.

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