- #1

Zatman

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**1. Homework Statement**Two particles of mass

*m*are connected by a light inextensible string of length

*l*. One of the particles moves on a smooth horizontal table in which there is a small hole. The string passes through the hole so that the second particle hangs vertically below the hole.

Derive the expression

[itex]\dot{r}^2 = A - \frac{B}{r^2} - gr[/itex]

where

*r*is the distance of the first particle from the hole, and

*A*and

*B*are constants. Use the equations of motion for each particle in polar coordinates.

## Homework Equations

General acceleration in polar coordinates:

[itex]\mathbf{\ddot{r}}=(\ddot{r}-r\dot{\theta}^2)\mathbf{\hat{r}}+(r\ddot{\theta}+2\dot{r}\dot{\theta}) \mathbf{ \hat{\theta} }[/itex]

## The Attempt at a Solution

I'm using a plane-polar coordinate system in the plane of the table with the origin at the hole. The forces acting on the first particle (on the table) (in the plane) are just the tension,

*T*. So:

[itex]-T\mathbf{\hat{r}} = (\ddot{r}-r\dot{\theta}^2)\mathbf{\hat{r}}+(r\ddot{\theta}+2\dot{r}\dot{\theta}) \mathbf{ \hat{\theta} }[/itex]

And for the hanging particle, the forces are the tension and weight, so (aligning the z axis perpendicular to the table):

[itex](T-mg)\mathbf{\hat{z}} = -m\ddot{r}\mathbf{\hat{z}}[/itex]

From these equations, I get the following system:

[itex]\ddot{r} - r\dot{\theta}^2 = -\frac{T}{m}[/itex]

[itex]r\ddot{\theta} + 2\dot{r}\dot{\theta} = 0[/itex]

[itex]\ddot{r} = g - \frac{T}{m}[/itex]

I've tried manipulating these equations for a while now but don't get anything close to the required form. Are these equations incorrect? I would be very grateful for any hints! :)