JEE Physics Oscillations Notes Set 02

Basics of Simple Harmonic Motion

1. SHM is motion in which the restoring force (or acceleration) is proportional to the displacement from the mean position and directed toward it: \( a=-\omega^{2}x \), \( F=-kx \).
2. Displacement \( x=A\sin(\omega t+\phi) \); velocity \( v=\omega\sqrt{A^{2}-x^{2}} \); maximum speed \( \omega A \) (at the mean position); maximum acceleration \( \omega^{2}A \) (at the extremes). Period \( T=\dfrac{2\pi}{\omega} \) and frequency \( f=\dfrac{1}{T} \).
3. If an equation has the form \( a=-\beta(x-x_0) \), the equilibrium position is \( x_0 \) and \( \omega=\sqrt{\beta} \); shifting the origin does not change the period.
4. A function such as \( \sin^{2}\omega t \) or \( \sin\omega t\cos\omega t \) is also SHM (about a shifted mean for the first, with double the frequency) because it can be written as a single sine or cosine.
5. From the extreme position to the mean position takes T/4. From the mean position to half the amplitude takes T/12, and from half the amplitude to the extreme takes T/6.

Energy in SHM

Kinetic energy \( \dfrac{1}{2}m\omega^{2}(A^{2}-x^{2}) \), potential energy \( \dfrac{1}{2}m\omega^{2}x^{2} \), total energy \( \dfrac{1}{2}m\omega^{2}A^{2} \), which stays constant. Kinetic and potential energies are equal at \( x=\pm\dfrac{A}{\sqrt2} \). Their averages over a period are each half of the total energy. At half the amplitude, potential energy is one quarter of the total.

Spring-Mass and Pendulum Systems

Damped Oscillations and Other Points

1. With damping, amplitude decays exponentially and the energy falls as \( E=E_0e^{-\gamma t} \); the initial phase is a property of the starting condition and is not changed by damping.
2. A platform oscillating vertically with amplitude A keeps an object on it only if \( \omega^{2}A\le g \).
3. For a pendulum whose bob is a hollow ball filled with water that leaks out, the period first rises and then falls back, because the centre of mass first goes down and then returns.

Worked Examples

1. A body in SHM has velocities 10 cm/s and 8 cm/s at displacements 4 cm and 5 cm. Then \( 100=\omega^{2}(A^{2}-16) \) and \( 64=\omega^{2}(A^{2}-25) \), giving \( \omega=2 \) rad/s and \( T=\pi \) s.
2. A spring-mass system with time period T has its mass increased by m so that the period becomes \( \dfrac{5T}{3} \). Then \( \dfrac{M+m}{M}=\dfrac{25}{9} \), so \( \dfrac{m}{M}=\dfrac{16}{9} \).
3. A mass 20 kg hangs on a spring of constant 2000 N/m and is released from rest when the extension is 15 cm. Equilibrium extension = 10 cm, amplitude = 5 cm and \( \omega=10 \) rad/s, so the displacement from equilibrium is \( y=5\cos10t \) cm.
4. A pendulum is lifted to a height of 64 km (R = 6400 km). The period increases by \( \dfrac{h}{R}=1\% \).