Physics Conical Pendulum

Consider a small spiracle body is attached at the lower end of string whose upper end is fixed when a small spiracle body is moved in horizontal circular the string attached to the body swipe over the surface of cone the whole system is called conical pendulum.
conical pendulum


Consider a small spiracle body of mass m is moving in a horizontal surface with angular velocity ω with center at o. at any moment the body at ‘a’ the string attached to body moves an angle θ with vertical the force are acting on the body are as shown in figure.


Since there is no acceleration in the vertical direction, the vertical component of the tension in the string is equal and opposite to the weight
T cosθ balance the  weight ‘mg’ of body
 T cos θ = mg  ----------------------(1)


T sin θ is acting towards the center of the circular path. This component will provide circular path.This component will provide the required centripetal force for circular motion.
T sin θ= mrω2   -----------------------(2)

Dividing equation (2)by equation (1)

T sin θ/ T cos θ = mr ω2/mg
Tan θ  = r ω2/g
ω= √g Tan θ /r -------------------------(3)

If T1 be the time period of revolution of the conical pendulum
T1 =  2 π/ ω
T1 =  2 π√r/g Tan θ
T1 =  2 π√ L sin θ/ g sin θ/cos θ
T1 =  2 π√ L sin θ/ g