As you know, when a reference particle revolves along the reference circle with a constant angular velocity, omega, the projection of the particle on the diameters along the X-axis and Y-axis executes simple harmonic motion.
The particle executes circular motion with a constant velocity ‘V’ whose magnitude is equal to ‘aω’ where ‘a’ is the radius of the reference circle.
For simplicity, if we assume the initial phase (
is equal to zero, then the two components of velocity V are V cos θ and V sin θ respectively in the directions shown.
Then, the component V sin θ represents the velocity of the projection, which is executing simple harmonic motion on the diameter along the X- axis.
Velocity of a particle in simple harmonic motion is equal to - V sin θ. The negative sign is due to the direction of the velocity at that instant which is opposite to that of the positive X-axis.
If the particle has an initial phase (
, the velocity of simple harmonic motion can be written as
Acceleration in simple harmonic motion
Acceleration of the revolving particle will be the centripetal acceleration
This centripetal acceleration has two components:
along the X-axis and
along the Y-axis.
If the initial phase Ф is equal to zero, the acceleration of the projection which is executing simple harmonic motion along the diameter along the X-axis is equal to
since this component is directed opposite to the positive X-axis.
If the initial phase phi is not equal to zero, the acceleration of the projection in simple harmonic motion is equal to
This can be written as
, which in turn can be written as
On rearranging, we get
The quantity a cos omega t plus phi is the displacement x of the particle executing simple harmonic motion.
Hence, the acceleration of the particle executing simple harmonic motion can be written in terms of its displacement as