Periodic and oscillatory motions
* Motion which repeats itself after a fixed interval of time is called periodic motion.
* The to-and-fro motion of a body is called oscillatory motion.
* All oscillatory motions are periodic motions. But every periodic motion need not be an oscillatory motion.
* Simple harmonic motion is the to-and-fro motion of body where the force is always directed towards the mean position and is proportional to the displacement but in the opposite direction.
Period, frequency and displacement
* Time period, T = Time taken/Number of Oscillations
* Frequency, n = Number of Oscillations/ Time taken
n=Number of oscillations/Time taken
n=1/T
* The displacement variable in general can be understood as the variations in physical quantity with time in a border sense when the variable involved oscillates.
* A linear combination of two periodic functions is also periodic.
* Any periodic function can be expressed as a superposition of sine and cosine functions with suitable coefficients.
Simple harmonic motion (SHM)
* A periodic motion where the displacement of an oscillating particle varies sinusoidally with time ‘t’ is called simple harmonic motion.
* The maximum displacement of a particle from the mean position executing simple harmonic motion is called its amplitude ‘A’.
| Time Independent |
Time Dependent |
| Amplitude | Displacement Velocity Acceleration (ωt +φ) |
This omega is called the angular frequency of the simple harmonic motion and its SI unit is radian per second.
Simple harmonic motion and uniform circular motion
* Since angular velocity is constant, the particle executes uniform circular motion.
* If the reference particle, P, executes uniform circular motion, its projection executes simple harmonic motion.
* If the revolving particle began its journey at the extremities of the diameter, then the projection on the diameters along the y-axis, its displacement will be zero and the initial phase angle phi would be zero.
Velocity and acceleration in simple harmonic motion
Velocity = – V sin(ωt +φ)
= – ⍺ω sin (ωt+φ)
Acceleration = -ω2x
Force law in simple harmonic motion
F(t) = – mω2x
F(t)= – kx
Angular frequency, ω = √k/m
* Linear harmonic oscillator : Force is directly proportional to displacement.
* Non linear harmonic oscillator: force is dependent on displacement but not exactly directly proportional
Energies in simple harmonic motion
Spring constant, k =mω2
Kinetic energy, K.E. = ½ ka2sin2(ωt)
Spring force, F = – kx
Potential energy, U(t) = ½ kx2
Total energy = ½ ka2
Spring and simple pendulum
Angular frequency, ω = √k/m
Time period, T = 2π √m/k
Angular acceleration, ⍺ = -mglsin⍬/I
Time period, T= 2π √l/g
Damped simple harmonic motion
* Mechanical energy is dissipated while doing work against the resisting force.
* The amplitude of the oscillations decreases with time and thus the oscillations of the pendulum are known as damped oscillations.
* Forces that resist the oscillations of a pendulum are known as damping forces.
* Damping forces, F_{1} ∝ -v
Forced oscillations
* The vibrations of a body can be of two types, namely free vibrations and forced vibrations.
* When a body is excited and released, the vibrations of the body are called free vibrations.
* When the body oscillates after an external force with angular frequency, ω0 is applied, the oscillations of the body are called ‘forced oscillations or driven oscillations’.
Amplitude, A = f0/m (ω2– ω02)
Resonance
Amplitude, A= f0/ω0b
* An increase in the amplitude of a body when the driving frequency is close or equal to its natural frequency is called resonance.
* There are many examples of resonance such as pendulums connected to the same rubber cord and the shattering of a wineglass by a performing singer.