**Motion of Object in a Horizontal Circle: Conical Pendulum**

A conical pendulum is the simple pendulum whirled in a horizontal circle such that the string makes a constant angle with the vertical and traces the surface of a right circular cone.

Consider an object of mass m tied to a string of length l and whirled in a horizontal circle of radius r with velocity v as shown in the figure. O is the point of suspension of the string and B is the center of the circular path. Let the string make an angle θ with the vertical. When the body is at point A, the tension T has two components Tsinθ and Tcosθ. The Tcosθ balances the weight mg of the object and Tsinθ provides the necessary centripetal force mv^{2}/r.

Tcosθ = mg ….... (i)

Tsinθ = mv

^{2}/ r …...... (ii)Dividing (ii) by (i), we get

Tanθ = v

^{2}/rg …..... (iii)From the figure, tanθ = r/ h …...... (iv)

From (iii) and (iv) we get

r/h = v

^{2}/rgor, r

^{2}/v^{2}= h/gso, r/v = √(h/g) ….... (v)

If t be the time period of oscillation of conical pendulum then

t = 2πr/v ….... (vi)

From equation (v) and (vi), we get

or, t = 2π √(h/g) ........ (vii)

But from the figure h = l cosθ

so, t = 2π √ (l cosθ/g) ......... (viii)

This is the expression for time period of oscillation of conical pendulum.

Squaring (i) and (ii) and adding we get,

T

^{2}= (mg)^{2}+ (mv^{2}/ r)^{2}or, T = mg √ [1 + (v

^{2}/ rg)^{2}] ….... (ix).: T = mg √ [1 + (r/ h)

^{2}] ….... (x)Equations (ix) and (x) give tension on the string.

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