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What is simple harmonic motion?

Class 11 PhysicsNCERT chapter: Oscillationsइसे हिंदी में पढ़ें

Exam-ready answer

Simple harmonic motion (SHM) is a periodic to-and-fro motion of a particle about a mean position in which the restoring force is directly proportional to the displacement from the mean position and is always directed towards it: F = −kx. The displacement is x(t) = A cos(ωt + φ), and the acceleration is a = −ω²x.

Explanation

A body oscillates when it moves back and forth about a mean (equilibrium) position. At the mean position the net force on it is zero. When displaced, a restoring force pulls it back. In SHM this force is linear: F = −kx, where k is the force constant and the minus sign shows the force points opposite to the displacement. A block attached to a spring on a smooth surface is the standard example.

The displacement is x(t) = A cos(ωt + φ). Here A is the amplitude (maximum displacement), ω is the angular frequency, and (ωt + φ) is the phase, with φ the phase constant. The motion repeats after a time period T = 2π/ω, and the frequency is ν = 1/T, measured in hertz. Differentiating twice gives the acceleration a = −ω²x, so acceleration is proportional to displacement and opposite to it.

Comparing F = ma = −mω²x with F = −kx gives ω = √(k/m) and T = 2π√(m/k) for a spring-block system. A simple pendulum makes SHM only for small angles of swing, with time period T = 2π√(L/g), where L is its length. Velocity is maximum at the mean position and zero at the extreme positions, while acceleration is zero at the mean position and maximum at the extremes.

Points to remember

  1. SHM: restoring force ∝ displacement and directed towards the mean position, F = −kx.
  2. Displacement: x(t) = A cos(ωt + φ); A = amplitude, ω = angular frequency, φ = phase constant.
  3. Acceleration: a = −ω²x.
  4. Time period T = 2π/ω; spring-block T = 2π√(m/k); simple pendulum T = 2π√(L/g) for small angles.
  5. Every SHM is periodic, but not every periodic motion is SHM.

Exam tip

Start with F = −kx and explain the minus sign. Then write x(t) = A cos(ωt + φ), name each symbol, and derive T = 2π√(m/k) by comparing with F = −mω²x.

Questions students also ask

What is the difference between periodic and oscillatory motion?
A motion that repeats itself after equal intervals of time is periodic, such as the Earth going round the Sun. Oscillatory motion is periodic motion back and forth about a mean position, such as a swing. Every oscillatory motion is periodic, but not every periodic motion is oscillatory.
Why is a simple pendulum in SHM only for small angles?
The restoring force on the bob is mg sin θ. For small angles, sin θ is nearly equal to θ, so the force becomes proportional to the displacement, which is the condition for SHM. For large angles this approximation fails.
Where is the speed maximum in SHM?
The speed is maximum at the mean position, where the displacement is zero, and its value is ωA. At the extreme positions the speed is zero because the particle momentarily stops before turning back.

Chapter labels help you find the topic in your textbook. Check your own edition: chapter names and order can change. These are Know.it explanations, not official NCERT solutions.