We know that when a body is set up in vibrations from its original/mean position by external force it come backs to its mean position due to action of restring force present in the body and said to be performing the linear simple harmonic motion. Whereas the linear motion of object along the circumference of circle with uniform velocity is known as uniform circular motion. The acceleration of particle in linear SHM is given as,
Where ω is angular velocity and x is displacement of particle.
Let’s understand the concept with appropriate formula……..!
Consider a particle performing UCM along circumference of circle of radius ‘a’ in anticlockwise sense as shown in fig.
Let ω be the constant angular velocity of particle. The particle is at point P1 on the circumference of the circle, its projection particle on the diameter AB at point O. The particle is moving along the circumference of circle and reach to point P2 in time t. Then the projection particle is at M on the diameter AB. The angular displacement of particle is,
∴ θ = ωt + α
where α is initial phase angle of particle
Draw perpendicular from P2 on AM at M, we get
From ∆ OP2 M,
Sin ωt + α = x/a
∴ x =a sin (ωt+α) ……..(1)
This is an expression for an instantaneous displacement projection particle.
Now diff. equation (1) w.r.t. ‘t’ we get,
dx/dt = a ω cos (ωt + α)
i.e. velocity, v = a ω cos (ωt + α) ……(2)
differentiating equation (2) w.r.t. ‘t’ again we get,
From the above equation it is clear that acceleration of particle performing SHM is directly proportional to the displacement and both are oppositely directed. Thus the motion described by the particle is linear SHM.
Hence it can be proven that the projection of UCM along any diameter is SHM.