# Rotating table with a groove having a ball within

A circular table of radius R is rotating with an angular speed Ω. It contains a groove of lenth L starting from the center of the the table and running along the radius vector of the table. A ball of diameter as same as the width of groove is kept in the groove, with initial velocity 0 ,at a distance a from the centre of the table. How can you describe the "motion" and the "forces" acting on the ball as when observed from an inertial frame? Please help me with this.

Edits:

What i think is,As considered from a non inertial frame,consider rotating coordinate axes rotating with the same angular velocity as that of the table.Let the ball be moving along the +ve x axis. Now, the weight (mg) (along -ve Y Axis) will be cancelled out by the normal reaction force N3 ( Along +ve Y Axis ). The normal reaction forces along the N2 is cancelled out by the N1 and the Coriolis force. The ball is then accelerated along the +ve X-Axis axis by the centrifugal force.

But now, when i consider from an inertial frame, there is no Coriolis force, hence, N2 dominates over N1 ( but i cant find any tangential acceleration since the table is moving with a uniform angular velocity) and i cant find any forces which can give the ball an outward acceleration(as it has to be moving outward) or any centripetal force( as it should be changing its direction along with the groove).

I find myself in total confusion!!

Use the equations of motion in polar coordinates to describe the motion of the ball.

\begin{align} F_r & = m ( \ddot{r} - r \dot{\theta}^2 ) \\ F_t & = m (r \ddot{\theta} + 2 r \dot{r} \dot{\theta} ) \end{align}

For your problem you will observe that the disk is spinning at a constant rate, so $\dot{\theta} = \Omega$ and $\ddot{\theta}=0$. Also you will see the ball is free to move along the groove and hence $F_r = 0$.

Use the above equations for solve for the radial acceleration $\ddot{r}$ and the groove reaction force $F_t$.

• Sorry., i dont get that in the polar coordinates. Can you explain it in general coordinate system? Commented May 13, 2017 at 17:02
• Did you read the wikipedia article which is linked. I can't fathom describing this better than wikipedia. Commented May 13, 2017 at 19:18
• I get what you have given. Please check the questions edit. "when i consider from an inertial frame, there is no Coriolis force, hence, N2 dominates over N1 ( but i cant find any tangential acceleration since the table is moving with a uniform angular velocity) and i cant find any forces which can give the ball an outward acceleration(as it has to be moving outward) or any centripetal force( as it should be changing its direction along with the groove). " Commented May 14, 2017 at 10:05
• @ja72 the ball is free to move along the groove so why $F_r = 0$ ??. I think if something is free to move in a direction, that doesn't mean the acceleration of that direction is 0
– Dat
Commented Nov 22, 2018 at 12:12
• Actually the opposite. Forces are there to enforce constraints, i.e. to make things move in a path they don't want to naturally follow. If it is free to move radially, it means there is no force in that direction. If there was a force then that force would produce work (force times distance) and you would generate energy out of nothing. Commented Nov 22, 2018 at 21:15