Name: Damped Harmonic Oscillations
Uploaded: 2019-07-19
Duration: 9 min 23 s
Description: An object floating in the water, when pushed slightly, starts oscillation but comes to rest after some time. This is the concept of damped oscillation. Let's...

Question

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

Accepted Answer

An S.H.M. or simple harmonic motion is the periodic to and fro motion of particle about its mean position. In this motion a restoring force is applied that is proportional to the displacement and is in a direction which is opposite that of the displacement.

To study the motion of a particle performing circular motion, let's consider a particle P moves on the circumference of circle of radius $A$ with uniform acceleration. Let PN is the perpendicular drawn from P to diameter $y y'$ .

From the given figure, $∠ P O X = θ$ , $O P = A$ and $O N = y$
From the right-angled triangle $∆ O N P$ ,

$\sin θ = \frac{O N}{O P}$

Since, $θ = ω t$

$\Rightarrow O N = O P \sin (ω t)$

Substituting the values from above,

$y = A \sin (ω t)$ ----(1)

This is the expression of equation of SHM.

Since $v = \frac{d y}{d t}$
Differentiate equation (1) with respect to time $(t)$ , to find velocity of particle.

$v = \frac{d (A \sin (ω t))}{d t}$

$v = A ω \cos (ω t)$ .....(2)

From the figure, $\cos (ω t) = \frac{\sqrt{A^{2} - y^{2}}}{A}$

$\Rightarrow v = ω \sqrt{A^{2} - y^{2}}$

This is the expression of velocity of particle in SHM.

Acceleration of the particle $a = \frac{d v}{d t}$

Differentiate equation (2) with respect to time $(t)$ ,

$a = \frac{d (A ω \cos (ω t))}{d t}$

$a = - ω^{2} A \sin (ω t)$

From equation (1),

$\Rightarrow a = - ω^{2} y$
This is the expression of acceleration of particle in SHM.

From the above expression of acceleration, we can see that the acceleration of particle is directly proportional to displacement and directed is in opposite direction and the same is true in case of SHM.

Thus, it can be said that motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic.

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

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