Q.Define simple harmonic motion. Show that the motion of (point) projection of a particle performing uniform circular motion on any diameter is simple harmonic.
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Start your 14-day free trial to unlock the full solution →SHM is a to-and-fro motion where a = -(omega^2)x. The projection of a particle in uniform circular motion onto a diameter executes x = A cos(omega t), giving a = -(omega^2)x, so it performs SHM.
Definition of SHM: Simple harmonic motion is the oscillatory (to-and-fro) motion of a body in which the acceleration (or restoring force) is directly proportional to the displacement from a fixed mean position and is always directed towards that mean position.
Mathematically: a = -(omega^2) x, where omega is the angular frequency and the minus sign shows the acceleration is opposite to the displacement.
Projection of uniform circular motion is SHM:
Consider a particle P moving with uniform speed on a circle of radius A and centre O, with constant angular velocity omega. Suppose it starts on the reference (x) axis and rotates anticlockwise. After time t, the radius OP makes an angle theta = omega t with the x-axis.
Let N be the foot of the perpendicular drawn from P onto a diameter (say the x-axis). As P moves round the circle, N moves back and forth along the diameter between its two ends. N is the projection of P on the diameter.
Displacement of N from the centre O:
x = A cos(theta) = A cos(omega t)
Velocity of N (differentiate x with respect to time):
v = dx/dt = - A omega sin(omega t)
Acceleration of N (differentiate v):
a = dv/dt = - A omega^2 cos(omega t)
Since A cos(omega t) = x, this becomes:
a = - omega^2 x
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