Where does an oscillator store its energy?
Where does an oscillator store its energy?
- Pause at a turning point and identify the energy store.
- Double stiffness while keeping mass fixed.
- Double amplitude and compare total energy.
What to measure
k in N/m; m in kg; A and x in m; E in J.
The idea underneath
A Hookean spring exerts a force proportional and opposite to displacement. With no damping the mass executes a sinusoid. Potential energy peaks at the turning points, where speed is zero; kinetic energy peaks at equilibrium, where spring force is zero. Zero force at one instant does not mean zero velocity. Changing stiffness changes the time scale, while changing amplitude changes energy quadratically.
x=A cos(ωt) · ω=√(k/m) · E=½kA²k in N/m; m in kg; A and x in m; E in J.
Build the argument
- Draw the forces and choose coordinates before writing a differential equation.
- Check the result against energy conservation and dimensional consistency.
- Change one parameter at a time and compare a ratio rather than relying on visual impressions.
Worked example 1
k=12 N/m, m=1 kg gives T=1.814 s.
Worked example 2
A=0.8 m gives E=3.84 J. At A=0.4 m, E=0.96 J.
Applications
Oscillators appear in clocks, vibration isolation, sensors and mechanical structures. Conserved quantities make complicated motion easier to predict.
Common misconception
A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.
A further investigation
Choose a mass that halves the frequency.
Check your understanding
At equilibrium the kinetic energy is…
- Maximum
- Zero
- Negative
Explanation
At x=0 the spring stores no potential energy, so all the energy is kinetic. Four times the mass halves frequency.
Model limits
Ideal linear spring, no damping or gravity along the horizontal track.