Motion of Object in a Horizontal Circle (Conical Pendulum) | Physics Grade XI

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Motion of Object in a Horizontal Circle (Conical Pendulum) | Physics Grade XI

Motion of Object in a Horizontal Circle (Conical Pendulum)

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.

Motion of object in a horizontal circleConsider 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 mv2/r.

Tcosθ = mg ….... (i)
Tsinθ = mv2/ r …...... (ii)
 
Dividing (ii) by (i), we get
Tanθ = v2/rg …..... (iii)
 
From the figure, tanθ = r/ h …...... (iv)
 
From (iii) and (iv) we get
r/h = v2/rg
or, r2/v2 = h/g
so, 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,
T2 = (mg)2 + (mv2/ r)2
or, T = mg √ [1 + (v2/ rg)2] ….... (ix)
.: T = mg √ [1 + (r/ h)2] ….... (x)
Equations (ix) and (x) give tension on the string. 

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