Circular motion

Their is two ways of measuring the speed of a body on a circular path, the strait line speed in the direction of the tangent, or the rate of change of the radius called angular speed given by: $$\omega = \frac{\theta}{t}$$ where $\omega$ is measured in radians per second.

Given an arc of radius $r$ and angle $\theta$ the arc length is $r\theta$ hence $v = \frac{r\theta}{t}$. Given $\omega = \frac{\theta}{t}$ we can derive the conversion formular: $$v = r \omega$$ Similarly the acceleration is given by: $$\begin{align} a &= r \omega^2\\ a &= \frac{v^2}{r} \end{align}$$

Example

A body of mass $800 \text{ kg}$ travels along a rough circular path of $40 \text{ m}$ with an angular speed of $0.2 \text{ rad s}^{-1}$. Find the least value of the coefficient of friction.

$$\begin{align} v &= r \omega \\ &= 40 \times 0.2\\ &= 8 \end{align}$$ $$\begin{align} a &= r\omega^2\\ &= 40 \times 0.2^2\\ &= 1.6 \end{align}$$ $$\begin{align} F &= ma\\ &= 800 \times 1.6\\ &= 1280 \end{align}$$ $$\begin{align} F \leq \mu R\\ 1280 \leq \mu \times 800 \times 9.8\\ \mu \geq 0.163 \end{align}$$