Class 11 Physics Oscillations Velocity in Simple Harmonic Motion

Velocity in Simple Harmonic Motion

  • Uniform Circular motion can be defined as motion of an object in a circle at a constant speed.
  • Consider a particle moving in circular path
  • The velocity at any point P at any time t will be tangential to the point P.
  • Consider θ = ωt+ φ where
  • θ = angular position
  • ω = angular velocity of the particle

    

  • To calculate the value of velocity along the x-axis for theprojectionP’which is executing SHM.
  • To find the component of velocity along -ive x-axis, we can write = -v cos(900 - θ)

This is can be written as:

  • = -v cos [900 – (ωt + φ)]
  • v (t) = - v sin(ωt+ φ) (Equation 1)

where

  • v (t) is instantaneous velocity of the particle executing the SHM.
  • (-ive sign tells that the velocity is directed towards negative x-axis)
  • To verify whether the Equation(1)is same if we calculate directly from SHM:-

x(t) = A cos(ωt + φ)

where

  • x(t) = displacement vector

v (t) = dx/dt

where

  • v(t) = velocity
  • dx/dt is rate of change of displacement

= -Awsin (wt + φ)

v (t) =  - A w sin(wt + φ)  (Same as Equation(1))

  • We can see from the above equation that the radius of the circular motionbecomes the amplitude of the SHM.

Instantaneous velocity of the particle executing SHM is given as:-  v (t) = - A w sin(wt + φ) 

In the above figure we can see that the velocity, v (t), of the particle P’ is the projection of the velocity v of the reference particle, P.

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