# Conical Pendulum & Time period equation – derivation | Problem solved

Last updated on February 12th, 2022 at 10:06 am

We will discuss the following topics (1) What is a conical pendulum? (2) the time period of the conical pendulum – equation or formula of time period (3) Derivation of the time period equation of the conical pendulum (4) Diagram of a conical pendulum (5) find out the equation of tension in the string of a conical pendulum (6) find out the semi-vertical angle of a conical pendulum

- What is a conical pendulum? | with diagram
- Tension in the string & length of the string (conical pendulum)
- Derive the Time period equation for Conical Pendulum
- Time period equation of conical pendulum
- How to find out the Tension in the string of a conical pendulum
- How to find out the semi-vertical angle θ of a conical pendulum

## What is a conical pendulum? | with diagram

A conical pendulum consists of a string OA whose upper-end O is fixed and a bob is tied at the free end.

Say, a horizontal push is given to the pendulum bob by drawing it aside.

And, say it describes a horizontal circle with uniform angular velocity ω in such a way that the string makes an angle θ with the vertical.

Then the string traces the surface of a cone of semi-vertical angle θ.

**It is called a conical pendulum. **In diagram (1) below, the angle AOH equals θ.

## Tension in the string & length of the string (conical pendulum)

Let us assume that T be the tension in the string, l be the length and r be the radius of the horizontal circle described.

Now, the vertical component of tension balances the weight, whereas the horizontal component supplies the centripetal force.

## Derive the Time period equation for Conical Pendulum

The angle AOH equals θ.

From diagram 1 above, Tcosθ = mg ; [vertical component of tension balances the weight]

Tsinθ = mrω^{2} [horizontal component of tension supplies the centripetal force]

∴ tanθ = (Tsinθ )/(Tcosθ) = rω^{2} / g

ω = (gtanθ / *r )*^{1/2} ………………… (1)

Again, r= lsinθ …………..(2)

Also, ω = 2π / t …………… (3) [**t is the time period, i.e., time for one revolution**.

We have used capital T as tension in the string, hence to avoid confusion we will use small t as the time period in the derivation part.]

from equation 1, 3 & 2 we get,

2π / t = (gtanθ / *r )*^{1/2}

=> 2π / t = (gtanθ / lsinθ* )*^{1/2}

t = 2π ( lcosθ/g ) ^{½}**t = 2π (h / g ) ^{1/2}** where h = l cosθ = height of the cone

Thus we have derived the equation of the Time Period of the conical pendulum as, **Time Period = 2π (h / g ) ^{1/2} **

## Time period equation of conical pendulum

**Time Period = T = 2π (h / g ) ^{1/2}**

## How to find out the Tension in the string of a conical pendulum

You can use either of the following equations to find out the value of tension T in the string. Pls, note that T denotes tension here.

– From diagram 1 above, T cosθ = mg ; [vertical component of tension balances the weight]

– And, T sinθ = mrω^{2} [horizontal component of tension supplies the centripetal force]

## How to find out the semi-vertical angle θ of a conical pendulum

Use any of the following equations according to the dependent data provided in the problem statement to find out the value of θ:

T cos θ = mg …… (a)

T sin θ = mrω^{2} … (b)

tan θ = (T sin θ )/(T cos θ) = rω^{2} / g

**θ = tan ^{-1} (rω^{2} / g)**…………… (c)