Complete ISC Class 11 Physics chapter on simple harmonic motion, damped and forced oscillations, sound waves, transverse and longitudinal waves, superposition, beats, resonance, and wave phenomena with mathematical derivations.
This chapter is central to understanding the physical world and forms the foundation for higher-level physics, engineering, and scientific reasoning.
Simple harmonic motion occurs when the restoring force is directly proportional to displacement and opposite in direction.
$$F = -kx$$
Hence,
$$a = -\frac{k}{m}x = -\omega^2 x$$
The general solution is
$$x = A \sin(\omega t + \phi)$$
where $A$ is amplitude and $\phi$ is the phase constant.
In SHM, total mechanical energy remains constant:
$$E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$$
At the mean position, kinetic energy is maximum and potential energy is minimum; at the extreme positions, the reverse is true.
For a simple pendulum, the restoring torque is proportional to angular displacement for small amplitudes, giving
$$T = 2\pi\sqrt{\frac{l}{g}}$$
This is valid for small angular displacements (less than about $10^\circ$).
Wave speed is related to frequency and wavelength by
$$v = f\lambda$$
In a transverse wave, particles oscillate perpendicular to the propagation direction; in a longitudinal wave, they oscillate parallel to it.
When two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements. This leads to interference and beats.
Beat frequency:
$$f_b = |f_1 - f_2|$$
Resonance occurs when a system is driven at its natural frequency, producing maximum amplitude. Standing waves are formed by the superposition of two waves of same frequency and amplitude traveling in opposite directions.
Nodes and antinodes are fixed positions of zero and maximum displacement, respectively.
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