xSkill: notes, lectures and practice for CBSE and CSE

1.1 The SHM equation Simple harmonic motion is a to and fro motion in which the restoring force is proportional to the displacement from the mean position and directed towards it. Writing the force as F = -k x gives m a = -k x, so the acceleration is proportional to the displacement and opposite to it. The solution of this equation is a cosine function of time, and the motion repeats after a fixed interval called the period.

x = A cos (ω t + ϕ), ω = \frac{2 π}{T} = 2 π f, a = - ω^{2} x

Displacement, angular frequency and the defining acceleration relation of SHM

1.2 Velocity and acceleration Differentiating the displacement once gives the velocity, and differentiating a second time gives the acceleration. At the centre of the motion the displacement is zero and the speed is at its largest; at the turning points the particle stops for an instant, so the speed is zero. The acceleration behaves the other way round: it vanishes at the centre and reaches its largest magnitude at the turning points. Since the acceleration always points back towards the centre, the particle slows down whenever it moves away from the centre and speeds up whenever it moves towards it. The largest speed is A omega at the centre, and the largest acceleration is omega squared A at the turning points, so both depend on the amplitude and on the angular frequency.

v = \frac{d x}{d t} = - A ω sin (ω t + ϕ), v = \pm ω A^{2} - x^{2}

Velocity in SHM in terms of time and in terms of displacement

1.3 Energy in SHM Because the force is conservative, the total mechanical energy of an oscillator stays constant. The kinetic energy is maximum at the mean position, the potential energy is maximum at the extremes, and the two trade places during the motion. The average kinetic energy and the average potential energy over a full cycle are each half of the total energy.

E = \frac{1}{2} k A^{2}, K = \frac{1}{2} m ω^{2} (A^{2} - x^{2}), U = \frac{1}{2} m ω^{2} x^{2}

Total, kinetic and potential energy of a simple harmonic oscillator

1.4 Spring and pendulum A mass on a spring and a simple pendulum are the two standard oscillators. For a spring the restoring force is provided by the spring itself, and the period depends on the mass and the spring constant. For a small-angle pendulum the restoring force is the component of weight along the arc, and the period depends on the length and on g, not on the mass of the bob.

T = 2 π \frac{m}{k} (spring), T = 2 π \frac{l}{g} (pendulum)

Period of a spring-mass system and of a simple pendulum for small oscillations

Exam

The pendulum period is independent of the mass of the bob and of the amplitude for small angles. Any question that makes the period depend on mass holds only for a spring, not for a pendulum.