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)
- 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)
- x(t) = displacement vector
v (t) = dx/dt
- 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.