Answer to Question #141543 in Mechanics | Relativity for Nama Glory

Question

Explain what is meant by angular speed. Derive an expression for the force required to make a particle of mass m move in a circle of radius r with uniform angular speed w.

Answer 1

Angular speed $\omega$ shows how fast an object rotates about a point (see https://en.wikipedia.org/wiki/Angular_velocity). It is measured in angle per unit time, the SI unit is 1/s.

$\omega = \dfrac{\Delta\varphi}{\Delta t}$ and if the point moves in a circle with radius r, the linear velocity is $v = \omega r.$

Let us calculate the centripetal force (see https://en.wikipedia.org/wiki/Centripetal_force) needed for body to rotate. "Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous centre of curvature of the path". For a small time interval t

$F_c = ma_c= m\cdot\dfrac{v}{ t} = m\cdot\dfrac{r\omega}{t} = m\cdot v \omega = m \cdot \omega^2 r.$