### Simple Pendulum

A particle of mass

*is attached by a light inextensible string of length*

**m***to a fixed point*

**l***It oscillates in a vertical plane under the force of gravity through a small angle .*

**O.**We have to fined the period of oscillation .

Let

*be the point of suspension,*

**O***the vertical .*

**OA**At any time

*let*

**t,***be the position of the particle,*

**P***being*

**Angle AOP***and*

**theta redians***being*

**arc AP**

**s.**The force that act on the particle are its weight

*vertically downwards and*

**mg***the tension of the string along*

**T**

**PO.**Resolving along the

*to the circle*

**tangent PQ**

**m(d/dt)(ds/dt)=-mg sin theta****------ (1)**

The amplitude of the oscillation is small and

therefore

**sin theta=theta**since ;

**s=theta l ,**equation

**(1)**reduces to

**(d/dt)(ds/dt)=gs/l****-------------- (2)**

Thus the motion is simple harmonic and the period of a

**Complete Oscillation**is

**2 pai multiplied by under root of (l/g)**
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