Answer to Question #154591 in Mechanics | Relativity for Sof

Question

Consider a circus act where the actor (of mass m) slides without friction under the

influence of gravity on a wire that makes an angle a with the upward verticle axis, 

and rotates with constant angular velocity w. 

(i) Write the Lagrangian using the appropriate generalized coordinates.

(ii) Write down the equation of motion for the generalized coordinates and discuss 

the motion of the stunt actor.

Answer 1

i) The Lagrangian:

$L=T-U$

where $T$ and $U$ are kinetic and potential energies of actor.

$T=\frac{1}{2}m(\dot{r}^2+r^2\omega^2 sin^2\alpha)$

where $r$ is a distance between fixed point and the actor.

$U=-mgcos\alpha$

$L=\frac{1}{2}m(\dot{r}^2+r^2\omega^2 sin^2\alpha)+mgcos\alpha$

ii) The equation of motion:

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{r}})-\frac{\partial L}{\partial r}=0$

$m\ddot{r}-mr\omega^2sin\alpha-mgcos\alpha=0$

$\ddot{r}-r\omega^2sin\alpha-gcos\alpha=0$

This equation shows that acceleration of actor is increasing when distance between fixed point and the actor is increasing. Also, acceleration is increasing when $\alpha$ or $\omega$ is incresing.