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2 Waves and Oscillations Schoology F3KC7-CQSXQ

3 Waves and Oscillations Displacement Equation of Oscillating Body Velocity and Acceleration of a body executing SHO Phase Time period and Frequency of Oscillation Graphical Representation of Oscillation Energy of Oscillating Body Average Energy of Oscillating Body Problems

Solution of the Differential Equation of SHM 4 Multiply both sides of eqn. by then we get Integrating with respect time, we have When y = a (amplitude) y y

Rearranging and the integrating we get Now putting the value of C we get Displacement equation of simple harmonic motion

6 y = a, v = min y = 0, v = max y The displacement of a particle executing simple harmonic motion is When y = o, v= max & When y=a, v=0 Waves and Oscillations Velocity of a body executing SHM

Problem:  The displacement of moving particle at any time t is given by. Show that the motion is simple harmonic.  An oscillatory motion of a body is represented by where symbols have their usual meaning. Show that the motion is symbol harmonic.  A body is vibrating with simple harmonic motion of amplitude 15cm and frequency 4Hz. Calculate the maximum value of velocity and the velocity when displacement is 9cm.

Confirmation  Write down the differential and displacement equation of simple harmonically oscillation body.  Calculate maximum and minimum velocity conditions for a particle executing simple harmonically.

HW The displacement of a simple harmonic oscillator is given by. If the oscillator started at time t = 0 from a position x 0 with velocity v = v 0, show that and

The maximum value of acceleration occurs at the position of one of its extreme displacements ( y = a ). Displacement equation y = a, v=0, a = max Acceleration of a body executing SHM Waves and Oscillations

Phase of a simple harmonically vibrating particle In the equation y = a sin (ωt + θ) the term (ωt + θ) = θ, is known as the phase.

12 Time period, frequency and angular frequency of a body executing SHM Time period: Frequency: Angular Frequency:

 A man stands on a platform which vibrates simple harmonically in a vertical direction at a frequency of 5 Hz. Show that the mass loses contact with the platform when the displacement exceeds meters.

Problem: The velocity of a particle executing simple harmonic motion are 4 cm s -1 and 3 cm s -1, when its distance from the mean position is 2 cm and 3 cm respectively. Calculate its amplitude, angular velocity and time period. Problem

Confirmation  Calculate maximum and minimum acceleration for a particle executing SHM.  Find an expression for the periodic time and frequency of a simple harmonic oscillator.  A mass m suspended from a spring of stiffness s executing SHM. Calculate its time period and frequency.

Energy of a body executing S.H.M. Potential energy of the particle Kinetic energy of the particle 17 Waves and Oscillations

Total energy of a particle executing SHM is proportional to the square of the amplitude of the motion. Total energy of the particle Waves and Oscillations Energy of a body executing S.H.M.

Waves and Oscillations Calculate the displacement to amplitude ratio for a simple harmonic motion when kinetic energy is 90% of total energy. Problem

Problem: A particle performs simple harmonic motion given by the equation. If the time period is 30 seconds and the particle has a displacement o f 10 cm at t = 0 seconds, find (i) epoch, (ii) the phase angle at t = 5 seconds.

Home Work  What is the ratio of kinetic energy at displacement one fourth to one third of the amplitude in case of simple harmonic motion?  A simple harmonic oscillator is characterized by Calculate the displacement at which kinetic energy is equal to its potential energy.

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