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

#EQUATION FOR TIME PERIOD FOR 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.

#VERTICAL COMPONENT

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)

#HORIZONTAL COMPONENT

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

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