Calculating the kinetic energy when angular velocity is involved

Question

Hi for the following question could someone please tell me what the coordinate transformations would be for example i think that we could use $$x=rsin\theta,y=rcos\theta,z=z$$ to determine the position since we know that $\theta=\omega t$, any help would be brilliant, its mainly part a forming the kinetic energy i know that $u=-mgz$

Answer 1

The question sets up a nice coordinate system for you. Let $x$ be position around the rotating axis (so $\frac{dx}{dt}$ is the velocity into the page) let $y$ be the distance from the rotating axis, and let $z$ be the height.

It's clear from the question that $y = r$, and $z = \alpha r$. So just differentiate them to get their velocity. $x$ is the centripetal speed, $\omega r$

The velocities are given by: $$\left( r\omega, \frac{dr}{dt}, \alpha\frac{dr}{dt} \right)$$ The magnitude of this, plugged into $E = 0.5 mv²$ should give the kinetic energy.