Simple Harmonic Motion (SHM) – Detailed Notes
1. Definition
Simple Harmonic Motion is a type of periodic motion in which a particle moves to and fro about
a fixed point such that the acceleration of the particle is directly proportional to its displacement
from the mean position and is always directed towards the mean position.
Mathematically:
a \propto -x
= acceleration
= displacement from mean position
Negative sign indicates acceleration is opposite to displacement.
2. Characteristics of SHM
1. Motion is oscillatory and periodic.
2. Restoring force is proportional to displacement from equilibrium.
3. Acceleration is always directed towards the mean position.
4. Velocity is maximum at the mean position and zero at extreme
positions.
5. Acceleration is maximum at extreme positions and zero at the mean
position.
3. Equation of SHM
Consider displacement at time :
x = A \sin(\omega t + \phi)
= amplitude (maximum displacement)
= angular frequency
= time period
= phase constant (initial condition)
Velocity:
v = \frac{dx}{dt} = A\omega \cos(\omega t + \phi)
Acceleration:
a = \frac{d^2x}{dt^2} = -A\omega^2 \sin(\omega t + \phi) = -\omega^2 x
4. Time Period, Frequency, and Angular Frequency
Time Period (T): Time taken for one complete oscillation.
Frequency (f): Number of oscillations per second, .
Angular Frequency (ω): .
5. Examples of SHM
1. Motion of a simple pendulum (small oscillations).
2. Vibration of a mass on a spring.
3. Oscillations of a tuning fork.
4. Motion of atoms in a crystal lattice.
5. Electrical oscillations in an LC circuit.
6. Energy in SHM
Total Energy in SHM is constant and is the sum of potential and kinetic energies.
Potential Energy:
PE = \frac{1}{2} k x^2
KE = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)
E = \frac{1}{2} m \omega^2 A^2 \quad (\text{constant})
7. Restoring Force in SHM
From Hooke’s law:
F = -kx
For a spring-mass system:
T = 2\pi \sqrt{\frac{m}{k}}
For a simple pendulum:
T = 2\pi \sqrt{\frac{l}{g}}
8. Graphical Representation
1. Displacement vs. Time → Sine or cosine curve.
2. Velocity vs. Time → Cosine or sine curve, phase-shifted by 90°.
3. Acceleration vs. Time → Sine or cosine curve, opposite in phase to
displacement.
9. Key Points to Remember
SHM is a special case of oscillatory motion.
Acceleration is proportional to displacement and directed towards the
mean position.
Total mechanical energy remains constant if no damping is present.
Time period depends only on system properties (mass, stiffness,
length, gravity) and not on amplitude (for ideal SHM).
