What is Simple Harmonic Motion?
Simple harmonic motion (SHM) is periodic back-and-forth motion where the restoring force is proportional to displacement and points toward equilibrium. Springs, pendulums and vibrating strings all approximate SHM.
In SHM the period of a mass-spring system is T = 2π√(m/k); the object oscillates with constant amplitude and frequency, governed by the restoring force F = −kx.
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Step-by-step worked examples
A 2 kg mass on a spring with k = 200 N/m. Find the period.
T = 2π√(m/k) T = 6.283×√(2/200) = 6.283×√0.01 T = 6.283×0.1 = 0.628 s
A 0.5 kg mass on a spring with k = 20 N/m. Find the period.
T = 6.283×√(0.5/20) = 6.283×√0.025 T = 6.283×0.1581 T ≈ 0.993 s
A 1 kg mass oscillates with period T = 2 s. Find the spring constant k.
T = 2π√(m/k) → k = 4π²m/T² k = 4×9.870×1/4 k ≈ 9.87 N/m
Flashcards
Quick quiz
Q1.What is the period formula for a mass-spring system?
Q2.If the mass is quadrupled and k stays the same, the period…
Q3.The restoring force in SHM always points…
Q4.For m = 2 kg and k = 200 N/m, what is the period?
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Common mistakes
Confusing period and frequency. — Correct: Period T is time per cycle (s); frequency f = 1/T is cycles per second (Hz).
Assuming a heavier mass always oscillates faster. — Correct: Heavier mass means a longer period (slower oscillation) for the same spring.
Ignoring the negative sign in F = −kx. — Correct: The sign shows the force always opposes displacement, pulling the mass back to equilibrium.
Thinking amplitude changes the period in ideal SHM. — Correct: For ideal SHM, period depends only on m and k — amplitude only affects how far the mass swings.
FAQ
What is simple harmonic motion?
Periodic oscillation where the restoring force is proportional to displacement, F = −kx, as seen in springs and pendulums.
What is the formula for simple harmonic motion period?
T = 2π√(m/k) for a mass-spring system, where m is mass and k is the spring constant.
What are examples of simple harmonic motion?
A mass on a spring, a swinging pendulum at small angles, and a vibrating guitar string.
How do you calculate the period of SHM?
Use T = 2π√(m/k) with the mass and spring constant, or T = 2π/ω if angular frequency is known.