Oscillations
Class 11 Physics · Physics Part II · Chapter 6
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Summary
A motion that repeats itself at regular intervals is periodic, and a to-and-fro motion about a mean position is oscillatory; every oscillation is periodic though not every periodic motion is oscillatory. The period \(T\) is the time for one cycle and the frequency \(\nu=1/T\) is measured in hertz. The simplest oscillation is simple harmonic motion (SHM), in which the displacement varies sinusoidally as \(x(t)=A\cos(\omega t+\phi)\), where \(A\) is the amplitude, \(\omega\) the angular frequency and \(\phi\) the phase constant. SHM is the projection of uniform circular motion on a diameter. Velocity \(v=-\omega A\sin(\omega t+\phi)\) and acceleration \(a=-\omega^{2}x\) are derived; acceleration is always directed toward the mean position and proportional to displacement. The force law \(F=-kx\) defines a linear harmonic oscillator with \(\omega=\sqrt{k/m}\). The kinetic and potential energies vary with time but their sum, the total energy \(E=\tfrac12 kA^{2}\), is constant. A simple pendulum executes SHM for small angles with period \(T=2\pi\sqrt{L/g}\), independent of mass and amplitude. These ideas underlie springs, pendulums and the vibrations of strings and air columns, and they form the basis for studying damped and forced oscillations and resonance.
Periodic and oscillatory motion; period and frequencySimple harmonic motionSHM and uniform circular motionVelocity and acceleration in SHMForce law for SHMEnergy in simple harmonic motionThe simple pendulum
Key terms
Periodic motion
Motion that repeats itself at regular intervals of time.
Simple harmonic motion
Oscillation in which displacement varies sinusoidally with time and the restoring force is proportional to displacement, \(F=-kx\).
Amplitude
The magnitude of the maximum displacement of a particle from its mean position in SHM.
Angular frequency
The quantity \(\omega=2\pi/T=2\pi\nu\) relating the period of SHM to its phase rate.
Phase
The argument \((\omega t+\phi)\) of the sinusoidal function that determines the state of the oscillator at any instant.
Restoring force
The force in SHM directed toward the mean position, proportional to and opposite to the displacement.
Extra questions & answers
Try one from this chapter
In simple harmonic motion the restoring force is: