Question

In simple harmonic motion, motion is executed by a particle that is subject to a force which is ________ to the displacement of the particle and is directed toward the _________.

• A. Proportional
• B. Mean position

Answer 1

Simple harmonic motion

1. Simple Harmonic Motion or SHM is an oscillatory motion in which the restoring force is directly proportional to the displacement of the body from its mean position.
2. The direction of this restoring force is always towards a fixed point(mean position).
3. The oscillation of a simple pendulum is an example of simple harmonic motion.

Mathematical view

Step 1: Velocity of the harmonic pendulum

Considering a simple harmonic wave $y=A\sin \omega t$

Where A is the amplitude, y is the displacement and $\omega$ is the angular velocity of the harmonic wave.

Differentiating y with respect to time, we get

Velocity, $\frac{dy}{dt}=\omega A\cos \omega t-----\left(1\right)$

Step 2: Acceleration of the harmonic pendulum

Again differentiate we get,

Acceleration, $\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}A\sin \omega t=-{\omega }^{2}y$, (since $y=A\sin \omega t$).

Or $\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}y$

or $a=-{\omega }^{2}y----------\left(2\right)$

The negative sign shows that the displacement and acceleration of the particle is directed to the opposite direction. i.e., the mean point

Step 3: Exerted force on the harmonic pendulum

From newtons law of motion, $F=ma$, where m and a are the mass and acceleration of the oscillator.

From equation $\left(2\right)$,

Force,

$$\begin{gathered} F=-m{\omega }^{2}y \\ F\propto y \end{gathered}$$

Force is directly proportional to displacement.

Therefore, it is clear that force is proportional to the displacement and acceleration (force) is directed towards the mean point.