Conical Pendulum | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

A conical pendulum

A is a ball on a string that moves in a horizontal circle.
The string length is $\ell$ . The string is at angle $\theta$ from the vertical.

The vertical part of tension balances weight. The inward part of tension turns the ball.

r=\ell\sin\theta

T\cos\theta=mg

T\sin\theta=\dfrac{mv^2}{r}

Your turnQuick check

Divide the inward equation by the vertical equation. Write the result without T or m.

Show worked answer

\tan\theta=\dfrac{v^2}{rg}

Worked example

Speed of a conical pendulum

A string is 0.80 m long and is at 30° from the vertical. Find the path radius and the speed of the ball.

Find the path radius.
$r=\ell\sin\theta$
$r=0.80\sin30^\circ=0.40\text{ m}$
Rearrange the result from the two force equations.
$v=\sqrt{rg\tan\theta}$
$v=1.50\text{ m s}^{-1}$

Answer

r=0.40\text{ m},\quad v=1.50\text{ m s}^{-1}

Common mistake

First find the line used to measure the angle. The sine and cosine components change when this line changes.