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.
Understanding Physics: Energy in Simple Harmonic Motion
The energy pattern follows from the shape of the spring energy curve. Stretching a spring twice as far requires more than twice as much stored energy. This happens because the spring force grows as the stretch grows.
Each small extra stretch needs work against a larger force than before. The result is a curved relationship. Position is represented by a parabola on an energy graph.
The curve has its lowest point at the centre and rises equally on either side. This symmetry explains why motion to the left and motion to the right have matching speeds at equal distances from the centre.
A useful way to track the motion is to divide one cycle into stages. At a turning point, the object pauses for an instant before reversing. Its acceleration is largest there because the spring pulls hardest there.
As it travels inward, the pull points in the same direction as its motion, so its speed increases. Halfway between the endpoint and the centre, neither type of energy has to be larger.
Their amounts are equal when the displacement has a particular value, about seven tenths of the amplitude. Near the centre, the spring pull becomes smaller, yet the object keeps moving quickly because it already gained kinetic energy on the way in.
Energy graphs need careful reading. The total energy graph is flat only in an ideal model where no energy leaves the moving system. The potential energy curve rises with the square of displacement.
The kinetic energy curve is an upside down version of that curve because both must add to the same total. A graph of position against time looks different. It is a smooth wave.
A graph of velocity against time is shifted by one quarter of a cycle. This shift means that when displacement is greatest, velocity is zero. Confusing these graph types is a common source of mistakes in tests.
Real oscillators gradually lose mechanical energy. Friction in a track, internal rubbing in a spring, and air resistance transfer energy to thermal energy and sound. The amplitude then becomes smaller after each cycle.
The period may remain nearly unchanged for light damping, but strong damping can prevent repeated oscillations. Car suspension systems use this effect deliberately. A spring stores energy after a bump, while a shock absorber removes energy so the car does not continue bouncing.
Pendulums, playground swings, guitar strings, and phone vibration motors show related energy transfers. When solving problems, first identify the amplitude and the chosen equilibrium position.
Then use conservation of energy only when losses are stated to be negligible. Mass changes the timing and the maximum speed, but for a given spring and amplitude it does not change the total stored energy.
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.