Question

A ball of mass 100 g is suspended a string 40 cm long. Keeping the string taut , the ball describes a horizontal circle of radius 10 cm Find the angular speed.

Answer 1

Given that,

Mass $m=100g$

Length $l=40cm$

Radius $r=10cm$

Now, by force equilibrium equation in vertical direction

$F_T\cos \theta =mg.....\left(I\right)$

By horizontal direction force equation

$F_T\sin \theta =m{\omega }^{2}r....\left(II\right)$

Now, the ratio of above equations

$\tan \theta =\frac{{\omega }^{2}r}{g}....\left(III\right)$

We know that,

$\sin \theta =\frac{1}{4}$

$\theta =14.47$

$\theta =14.5$

Now, from equation (III)

$\tan \theta =\frac{{\omega }^{2}r}{g}$

$\omega =\sqrt{\frac{g\tan \theta }{r}}$

$\omega =\sqrt{\frac{10\times \tan 14.5}{10}}$

$\omega =4.949$

$\omega =5.1rad/s$

Hence, the angular speed is 5.1 rad/s