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Class 10 Class 12

Oscillations

A particle is executing simple harmonic motion with amplitude r and frequency ω. Find the velocity and acceleration of particle at mean and extreme positions.

Given a particle is executing Simple Harmonic Motion.

The position, velocity and acceleration of particle executing simple harmonic motion is given by,

Position comma space straight y space equals space straight r space s i n space omega t V e l o c i t y comma space straight v space equals space straight omega square root of straight r squared minus straight y squared end root a n d space space space A c c e l e r a t i o n comma space straight a equals negative straight omega squared straight y

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Show that for a particle in linear simple harmonic motion, the average kinetic energy over a period of oscillation is half the total energy.

Let a particle of mass m execute simple harmonic motion with frequency ω. Let a be the amplitude of vibration.

The position of particle at any instant during simple harmonic motion is given by,

Now the average kinetic energy of particle over complete cycle is,

Hence proved.

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Derive the differential equation of simple harmonic motion.

The acceleration of particle executing simple harmonic motion, a = – ω²y

We know that the acceleration is second derivative of displacement.

Therefore,

$space space space space space space space space space space space space space space straight a equals fraction numerator straight d squared straight y over denominator dt squared end fraction therefore space space space fraction numerator straight d squared straight y over denominator dt squared end fraction equals negative straight omega squared straight y space space space space space space fraction numerator straight d squared straight y over denominator dt squared end fraction plus straight omega squared straight y equals 0$

The above equation represent the differential equation of simple harmonic motion.

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A particle is vibrating in a straight line with amplitude A. What is the total distance travelled and displacement of particle in one complete oscillation?

Let the particle be vibrating between points L and M. In one complete vibration the particle moves from O to L, then from L to O and then from O to M and finally from M to O as shown in figure. Therefore the total distance travelled by particle in one complete oscillation is

As the particle comes back to same position after one complete oscillation, therefore the net displacement of particle is zero.