Key ideas
A system in simple harmonic motion swings back and forth because a restoring force pulls it toward equilibrium in proportion to the displacement. The motion repeats with a fixed period.
- The period T is the time for one cycle and the frequency is f = 1 ÷ T. The angular frequency is ω = 2πf.
- For a mass on a spring T = 2π√(m ÷ k). For a small-angle pendulum T = 2π√(L ÷ g). Neither depends on the amplitude.
- The total energy ½kA² is constant and trades between kinetic and potential energy. Speed is greatest at the center and zero at the ends.
- Acceleration is greatest at the ends of the motion, where the restoring force is largest.
Key equations
T = 1 ÷ fω = 2πfT = 2π√(m ÷ k) (spring)T = 2π√(L ÷ g) (pendulum)E = ½kA²v_max = Aωa_max = Aω²
Worked example
A 0.50 kg mass on a 200 N/m spring has an amplitude of 0.10 m. ω = √(k ÷ m) = √400 = 20 rad/s, so T = 2π ÷ 20 = 0.31 s and f = 3.2 Hz. The maximum speed is Aω = 0.10 × 20 = 2.0 m/s. The energy is ½kA² = ½ × 200 × 0.010 = 1.0 J, which equals ½mv² at the center: ½ × 0.50 × 4.0 = 1.0 J.
Common mistakes
- Thinking a larger amplitude means a longer period. For ideal oscillators the period does not depend on amplitude.
- Believing a heavier pendulum bob swings more slowly. Its period depends on the length and on g, not on the mass.
- Mixing frequency f in hertz with angular frequency ω in radians per second.
Practice
Test yourself with practice questions on this topic: Oscillations