Question

Define simple harmonic motion. Show that the motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic.

Answer 1

simple harmonic motion : A body is said to be in simple harmonic motion when it moves to and fro along a straight line path about its mean position such that, at any point its acceleration is directly proportional to its displacement but in opposite direction and directed always towards the mean position.
e.g,. $a\propto-y$
where a denotes Acceleration and y denotes displacement of body in SHM.

Let's consider a particle P moves on the circumference of circle of radius A with uniform angular velocity $\omega$. Let particle P makes an angle $\theta$ in centre O after time t,
Now, Drawn a perpendicular line PN to y-axis. as shown in figure.
now, $sin\theta=\frac{ON}{OP}$
because we know, $\theta=\omega t$, ON = y (Let) and OP = radius of circle = A

so, $\boxed{y=Asin\omega t}$.......(1)

differentiate y with respect to t,
we get, $v=\omega Acos\omega t$
again, differentiate with respect to t,
we get, $\boxed{\bf{a=-\omega^2Asin\omega t}}$

from equation (1),
$\boxed{\bf{a=-\omega^2y}}$
hence, it is clear that acceleration of particle is directly proportional to displacement and directed in opposite direction, so motion is SHM. hence it is also clear that motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic.

Answer 2

Answer:hence, it is clear that acceleration of particle is directly proportional to displacement and directed in opposite direction, so motion is SHM. hence it is also clear that motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic

Explanation: