A 0.50 kg mass on a spring of spring constant 200 N m−1 oscillates with amplitude 4.0 cm. What is its maximum speed?
- ω = √(k/m) = √(200/0.50) = √400 = 20 rad s−1.
- Maximum speed = ωx0, with x0 = 0.040 m.
- = 20 × 0.040 = 0.80.
Answer: 0.80 m s−1
Oscillations
NUST NET (Engineering) · Physics · Oscillations, waves and light. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
The idea
In simple harmonic motion the acceleration is proportional to the displacement from the mean position and always points back towards it: a = −ω2x. The period does not depend on the amplitude.
At the mean position the speed is greatest and the acceleration is zero. At the ends the speed is zero and the acceleration is greatest. Energy swaps between kinetic and potential, while the total stays constant and is proportional to the amplitude squared.
Damping (friction) removes energy, so the amplitude dies away. Resonance happens when a driving force has the same frequency as the natural frequency of the system: the amplitude becomes very large.
Rules to remember
- a = −ω2x; ω = 2πf = 2π/T
- Mass on a spring: T = 2π√(m/k); for a load that stretches the spring by e, T = 2π√(e/g)
- Simple pendulum: T = 2π√(l/g), independent of the mass of the bob; a seconds pendulum has T = 2 s
- Starting from the mean position: x = x0 sin ωt; v = ω√(x02 − x2)
- Maximum speed = ωx0 (at the centre); maximum acceleration = ω2x0 (at the ends)
- Total energy = ½kx02 = ½mω2x02; PE = ½kx2; KE = ½k(x02 − x2)
- KE = PE when x = x0/√2; resonance when driving frequency = natural frequency
Common mistake
Treating the period as proportional to length or mass. T depends on the square root, so a pendulum four times as long has twice the period, and a heavier bob changes nothing.
Worked example
Practice questions
Try each one, then open the answer.
1. A load hung on a spring stretches it by 0.25 m. The load is pulled down a little and released. Taking g = π2 m s−2, the period of its vertical oscillations is
- A0.25 s
- B0.50 s
- C1.0 s
- D2.0 s
Show answer
Answer: C. At equilibrium kx = mg, so m/k = x/g. T = 2π√(m/k) = 2π√(0.25/π2) = 2π × 0.5/π = 1.0 s.
2. A body performs SHM with amplitude x0. When its displacement is x0/3, the fraction of its total energy that is kinetic is
- A1/9
- B2/3
- C8/9
- D1/3
Show answer
Answer: C. PE = ½kx2, which at x = x0/3 is (1/3)2 = 1/9 of the total ½kx02. So KE = 1 − 1/9 = 8/9 of the total. 1/9 is the potential-energy fraction, and 2/3 wrongly takes the energy ∝ x.
3. A washing machine shakes violently at one particular drum speed as it speeds up, then runs smoothly at higher speeds. The violent shaking happens because
- Aat that speed the drum turns at a natural frequency of the machine, so resonance occurs
- Bthe motor gives its greatest power at that speed
- Cthe damping of the machine becomes zero at that speed
- Dthe clothes are lightest at that speed
Show answer
Answer: A. The spinning drum is a periodic driving force. When its frequency matches a natural frequency of the machine body, energy is absorbed strongly and the amplitude becomes large: resonance. Above that speed the frequencies no longer match.
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