## Note on Simple Harmonic Motion in Terms of Uniform Circular Motion

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### 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 ymax = 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.

• Displacement of the 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*}
• The motion in which the position of a body repeats after fixed interval of time is known as periodic or harmonic motion.
• ,Amplitude is the maximum displacement of a body from its mean position in periodic motion.
• ,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.
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T/2

T/6

T/12

T/4

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• ### A hole is made through centre of earth and a stone is put into the hole. The morion of stone is ______.

stone stops at centre of earth

simple harmonic motion

linear

none of the answers are correct

• ### Bob of simple pendulum is fitted with liquid. If pluged hole at bottom of bob is suddendly unplugged then the time period of simple pendulum till water comes out ______.

first decreases and increases

first increases then decreases

increases

decreases

2n

n

n2

n/2

• ### Second's Pendulum id defined as

A simple pendulum having both kinectic energy and potential energy.
A simple pendulum whose time period is two seconds.
A simple pendulum whose frequency is two.
A simple pendulum whose phase difference is two.

4

3

1

2

1/2

3/2

1

• ### The string the spring and the pulley shown in figure are light. Find the time period of the mass m ______.

$$4pi sqrt {frac mk}$$

$$2pi sqrt {frac mk}$$

$$pi sqrt {frac mk}$$

$$2pi sqrt {frac km}$$

2

3

1

4

• ### What is time period of oscillation of liquid in U-shaped tube as shown in the figure?

$$pi ext {sec}$$

$$frac {pi}{2} ext {sec}$$

$$pi ext {sec}$$

$$2pi ext {sec}$$

• ### Find the time period of small back and forth on a smooth concave surface of radius R.

$$2pi sqrt {frac Rg}$$

$$2pi sqrt {frac gR}$$

$$pi sqrt {frac Rg}$$

$$pi sqrt {frac gR}$$

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