# Mechanics Circular Motions

Mechanics Circular Motions are handled by resolving forces horizontally and vertically in conjunction with the tension of the string, normal reactions, circular motion, and the particle’s mass.

### Worked Examples of Mechanics Circular Motions

A particle of mass $m$ is attached to one end of a string of length $R$. The other end of the string is fixed at height $2h$ above the centre of a sphere of radius $R$. The particle moves in a circle of radius $r$ on the surface of the sphere and has constant angular velocity $\omega > 0$. The string makes an angle of $\theta$ with the vertical.
Three forces act on the particle: the tension force $F$ of the string, the normal reaction force $N$ to the surface of the sphere, and the gravitational force $mg$. (a)    By resolving the forces horizontally and vertically, show that
$$F \sin \theta – N \sin \theta = m \omega ^2 r \\ F \cos \theta + N \cos \theta = mg$$.

(b)    Show that $\displaystyle N = \frac{1}{2} mg \sec \theta – \frac{1}{2} m \omega r \ \text{cosec} \ \theta$.

(c)    Show that the particle remains in contact with the sphere if $\displaystyle \omega \le \sqrt{\frac{g}{h}}$.