# Optional Practical: The conical pendulum

# The Conical pendulum

#### Introduction

A conical pendulum is a pendulum that is spun round in a circle instead of swung backwards and forwards. In this experiment a mass is attached to a string and made to spin in a circle of fixed radius, the time period of the motion is related to the length of the string. By varying the length and measuring Time period the acceleration of gravity can be found.

#### Research question

*How does the time period of a conical pendulum depend upon the length of the string?*

**Independent variable**: Length of pendulum

**Dependent variable**: Time period

**Controlled variables**: Radius, mass and properties of the string.

#### Method

Draw a clear circle of about 10cm diameter on a piece of paper. Holding the string above the centre of the circle swing the mass so it travels around the circumference. This isn’t that easy so it might be worth practising with different lengths before you start the measurements. You will notice that the time period is longer when the string is longer.

#### Theory

Above is a free body diagram for the mass. If the forces are resolved vertically and horizontally:

Vertically the forces are balanced so mg=Fcosθ

Where F is the tension

Horizontally there is centripetal acceleration so mω^{2}r=Fsinθ

Where

m = mass

ω = Angular velocity

r = radius

Dividing gives

mg/mω^{2}r =Fcosθ/Fsinθ =1/ tanθ

But

tanθ = r/h so ω^{2}r/g= r/h

from the definition

ω = 2π/T so 4π^{2}/gT^{2 }= 1/h

Where T = time period

Squaring gives

h^{2} = g^{2}T^{4}/16π^{4 }

From Pythagoras

L^{2} = h^{2} + r^{2} so h^{2}=L^{2} - r^{2}

Finally

L^{2}=g^{2}T^{4}/16π^{4} + r^{2}

By changing L and measuring T keeping r constant use a graphical method to find g