Simple Harmonic Motion in Terms of Uniform Circular Motion

Simple Harmonic Motion

The motion in which the position of a body repeats after fixed interval of time is known as periodic or harmonic motion. A harmonic motion is simplest type i.e. constant amplitude and simple frequency is known as simple harmonic motion. A body moves to and fro about it’s mean position, the acceleration so produced is directly proportional to the displacement 9y0 and is always directed towards the mean position.

$$\text {i.e.} a = -ky $$

Where negative sign shows that the acceleration and displacement are in opposite directions.

S.H.M. in Terms of Uniform Circular Motion

Let us consider a particle moving around a circle of radius r with a uniform angular velocity () in anticlockwise direction as shown in the figure. XOX’ and YOY’ are two mutually perpendicular diameters of the circle. As the particle goes in the circle, the foot of the perpendicular along Y-axis executes oscillatory motion.

Let at any time t, the position of the particle is P and angular displacement Ï´. Let M and N be the foot of the perpendicular drawn from P on XOX’ and YOY’ respectively. The displacement of the foot of perpendicular N on diameter YOY’ is ON.

\begin{align*} \sin \theta &= \frac {ON}{OP} \\ As \:ON &= y, OP = r \\ \therefore \sin \theta &= \frac yr \\ \text {or,} \: y &= r\sin \theta \\ \text {or,} \: y &= r\sin \omega t \dots (i) \end{align*}

Where \(\theta = \omega t\). This is the displacement equation for s S.H.M which is periodic, sinusoidal function of time. It can be expressed in terms of cosine function and similar expression can be obtained in any diameter of the circle.

Velocity

It is the rate of change of displacement of a body.

\begin{align*} v &= \frac {dy}{dt} = \frac {d(r\sin \omega t)}{dt} \\ &= r \omega \cos \omega t \dots (ii) \\ &= r\omega \sqrt {1 - \sin ^2 \omega t} \\ &= r\omega \sqrt { 1 - \left (\frac yr \right ) ^2} \\ &= \omega \sqrt {r^2 –y^2} \dots (iii) \end{align*}

$$ \text {Case I, if}\: y = 0 , V_{max} = \omega r$$

$$\text {Case II, If} \: y = r, V_{min} = 0$$

Acceleration

It is the rate of change of velocity of a body.

\begin{align*} a &= \frac {dv}{dt} = r\omega \frac {d}{dt} \cos \omega t \\ &= r\omega \times -\omega \sin \omega t \\ &= \omega ^2 r \sin \omega t \dots (iv) \\ &= -\omega ^2y\dots (v) \\ \end{align*}

Here \(a \propto y\) and is directed toward mean position so the motion is S.H.M.

Amplitude

It is the maximum displacement of a body from its mean position in periodic motion. The displacement in S.H.M. at any time is given by the relation, \( y = r \sin \omega t\). When \(\sin \omega t = 1\), then y_max = r. So, the amplitude of motion is r.

Time Period (T)

It is the time required by a body to complete one revolution.

$$ T = \frac {2\pi }{\omega } = 2\pi \sqrt {\frac ya} (\therefore a = \omega ^2y, \omega = \sqrt {\frac ay} )$$

Frequency (F)

It is the number of complete rotations made by a body in 1 second.

$$ F = \frac 1T = \frac {\omega}{2\pi} = 2\pi \sqrt {\frac ay} $$

Phase (Θ)

The phase of the body at any time is defined as the position and direction of its motion with respect to mean position at that time.