# 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

$F \sin \theta-N \sin \theta = m \omega ^2 r \\ F \cos \theta + N \cos \theta = mg$

Resolve forces for Mechanics Circular Motions.
\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} F \sin \theta &= N \cos \theta + m \omega r^2 r &\color{red} \text{horizontally} \\ \therefore F \sin \theta-N \sin \theta &= m \omega ^2 r &\color{red} (1) \\ F \sin \theta + N \sin \theta &= mg &\color{red} \text{vertically} \\ \therefore F \cos \theta + N \cos \theta &= mg &\color{red} (2) \end{aligned}

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

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \frac{F \sin \theta}{\sin \theta}-\frac{N \sin \theta}{\sin \theta} &= \frac{m \omega ^2 r}{\sin \theta} &\color{red} (1) \div \sin \theta \\ F-N &= m \omega ^2 r \ \text{cosec} \ \theta &\color{red} (3) \\ \frac{F \cos \theta}{\cos \theta} + \frac{N \cos \theta}{\cos \theta} &= \frac{mg}{\cos \theta} &\color{red} (2) \div \cos \theta \\ F + N &= mg \sec \theta &\color{red} (4) \\ 2N &= mg \sec \theta-m \omega^2 r \ \text{cosec} \ \theta &\color{red} (4)-(3) \\ \therefore N &= \frac{1}{2} mg \sec \theta-\frac{1}{2} m \omega^2 r \ \text{cosec} \ \theta \end{aligned}

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

$\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}}$
The particle stays in contact with the sphere $N \ge 0$.
\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \frac{1}{2} mg \sec \theta-\frac{1}{2} m \omega^2 r \ \text{cosec} \ \theta &\ge 0 \\ mg \sec \theta-m \omega^2 r \ \text{cosec} \ \theta &\ge 0 \\ – m \omega^2 r \ \text{cosec} \ \theta &\ge-mg \sec \theta \\ m \omega^2 r \ \text{cosec} \ \theta &\le mg \sec \theta \\ \omega^2 &\le \ddfrac{mg \sec \theta}{mr \ \text{cosec} \ \theta} \\ \omega^2 &\le \frac{mg \times \ddfrac{R}{h}}{mr \times \ddfrac{R}{r}} &\color{red} \sec\theta = \frac{R}{h}, \text{cosec} \ \theta = \frac{R}{r}\\ \omega^2 &\le \frac{g}{h} \\ \therefore \omega &\le \sqrt{\frac{g}{h}} \end{aligned}

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