In simple harmonic motion, a system such as a block on a spring repeatedly moves back and forth about an equilibrium position. Energy is the easiest way to see what is happening during the motion because it shows how speed, position, and force are connected. As the block moves, kinetic energy and elastic potential energy trade places continuously.
If friction and air resistance are ignored, the total mechanical energy stays constant throughout the cycle.
For a spring oscillator, elastic potential energy is greatest at the turning points x = +A and x = -A, where the spring is most stretched or compressed and the speed is zero. Kinetic energy is greatest at equilibrium x = 0, where the force is zero but the speed is maximum. The restoring force F = -kx always points toward equilibrium, so it slows the mass as it moves away from the center and speeds it up as it returns.
Energy graphs show this clearly: U changes with x squared, K fills in the difference, and E total remains a horizontal line.
Key Facts
- Hooke's law for a spring oscillator: F = -kx.
- Elastic potential energy: U = 1/2 kx^2.
- Total mechanical energy in ideal SHM: E = 1/2 kA^2.
- Kinetic energy at position x: K = 1/2 k(A^2 - x^2).
- Maximum speed occurs at equilibrium: vmax = Aω = A√(k/m).
- Maximum force occurs at the endpoints: Fmax = kA.
Vocabulary
- Simple harmonic motion
- Simple harmonic motion is repeated motion in which the restoring force is proportional to displacement and points toward equilibrium.
- Amplitude
- Amplitude is the maximum displacement from equilibrium, usually represented by A.
- Equilibrium position
- The equilibrium position is the central point where the net restoring force on the oscillator is zero.
- Elastic potential energy
- Elastic potential energy is energy stored in a stretched or compressed spring.
- Total mechanical energy
- Total mechanical energy is the sum of kinetic energy and potential energy in a system.
Common Mistakes to Avoid
- Saying the total energy is zero at equilibrium is wrong because kinetic energy is maximum there even though spring potential energy is zero.
- Putting maximum speed at x = ±A is wrong because the mass momentarily stops at the turning points before reversing direction.
- Treating force and velocity as always pointing in the same direction is wrong because the restoring force always points toward equilibrium while velocity depends on the direction of travel.
- Using U = kx^2 instead of U = 1/2 kx^2 is wrong because the factor 1/2 comes from the spring force increasing gradually with displacement.
Practice Questions
- 1 A 0.50 kg block is attached to a spring with k = 200 N/m and amplitude A = 0.10 m. Find the total mechanical energy of the oscillator.
- 2 For a spring with k = 80 N/m and amplitude A = 0.25 m, find the kinetic energy when the block is at x = 0.15 m.
- 3 Explain why the acceleration is largest at the endpoints even though the speed is zero there.