Oscillations By M P Chaphekar Types Of Motion

Oscillations • By M. P. Chaphekar

Types Of Motion 1. Translational Motion 2. Rotational Motion 3. Oscillatory Motion

Linear simple harmonic Oscillator Example of spring mass oscillator Displacement of Mass towards right Restoring force directed towards left Mass in equilibrium position . . Displacement of Mass towards left Restoring force directed towards right Restoring force F=-kx Restoring force directed towards mean position

Linear S. H. M. • Linear S. H. M. is defined as the linear periodic motion of a body , in which the restoring force (or acceleration ) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.

Differential Equation of linear S. H. M. Restoring force = force constant

Acceleration , velocity and displacement in S. H. M 1. Acceleration

Acceleration , velocity and displacement in S. H. M 2. Velocity

Acceleration , velocity and displacement in S. H. M 2. Velocity C is constant of integration At Hence

Acceleration , velocity and displacement in S. H. M 3 Displacement

Acceleration , velocity and displacement in S. H. M 3. Displacement is called phase of S. H. M.

How to understand (starting phase or epoch)? α Time Displacement Oscillations starts at t=0 X=0 Mean position and particle 0 moves towards positive extreme position t= 0 X=ɑ Positive extreme position t=0 X=0 Mean position and particle π moves towards negative extreme position t=0 X=-ɑ Negative extreme position 3π/2 t=0 X=0 Mean position 0 π/2

Kinetic energy and potential energy of a particle performing S. H. M 1. Kinetic energy OR

Kinetic energy and potential energy of a particle performing S. H. M 2 Potential energy Work done dw during the displacement dx is Work done to displace particle from 0 to x is stored as potential energy

Kinetic energy and potential energy of a particle performing S. H. M 3 Total energy Note– Total energy is independent of displacement x. Total energy depends upon mass , frequency of oscillation and amplitude of oscillation ,

Kinetic energy and potential energy of a particle performing S. H. M 4. Variation of K. E. and P. E. in S. H. M. Energy E P. E At mean position P. E. =0 K. E. =T. E At extreme position Displacement x K. E. =0 P. E. =T. E Note– Total energy is at any position in S. H. M. is constant i. e. conserved K. E

Graphical Representation of S. H. M 1 Particle performing S. H. M. Starting from mean position

Simple pendulum • Definition-An ideal simple pendulum is defined as a heavy particle suspended by weightless inextensible and twistless string from a rigid support.

2. Motion of simple pendulum as S. H. M Fig shows a simple pendulum of length L with a bob of mass m. The restoring force is If the is small, T L m mgsin (AS L m, g are consants) , Thus simple pendulum performs S. H. M Note mg mgcos

Motion of simple pendulum as S. H. M Period of simple pendulum T L m mgsin , mg mgcos

Damped Oscillations This is differential equation of damped harmonic oscillator.

Damped Oscillations The solution of the differential equation gives Graph of displacement against time Note: 1)Amplitude decreases with time exponentially 2)Period of oscillation increases.