1
$\begingroup$

A cylinder of radius $a$ is fixed in space. A hoop of mass $m$ and radius $b$ is in contact with the surface of the cylinder and it is able to rotate around of it without slipping. The force of gravity is acting on the system (figure below).

At $t=0$, the hoop is at rest and a tangential velocity $v_0$ is imposed in the lowest point of the hoop.

¿What must be the minimum magnitude of $v_0$ in order to make the hoop perform a complete loop around the cylinder?

enter image description here


I have no idea what condition must be imposed in order to describe the complete loop around the cylinder and then computing $v_0$. I think it has something to do with the rolling constraint and that the center of mass (COM) of the hoop must describe a complete arc lenght $2\pi(b-a)$.

The initial condition must be $\dot{\alpha}(t=0)=v_0/b$ (it comes from $\boldsymbol{v}_\perp=\boldsymbol{\omega}\times\boldsymbol{r}$), where $\alpha$ is the angle between a vertical axis fixed at the COM of the cylinder and a vertical axis at the COM of the cylinder.

$\endgroup$
1
$\begingroup$

This is similar to the question asking how fast a simple pendulum with a string must be launched in order to reach the highest point without the string becoming slack. At the highest point, gravity must be matched by the centripetal force - the tension in the string must not fall below zero. Here the equivalent of tension is the normal reaction from the cylinder.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.