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GeneralOJB

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## Homework Statement

The diagram shows a smooth thin tube through which passes a string with masses [itex]m[/itex] and [itex]M[/itex] attached to its ends. The tube is moved so that the mass [itex]m[/itex] travels in a horizontal circle of radius [itex]r[/itex] at constant speed [itex]v[/itex]

http://quickpic.info/z/yb.jpg [Broken]

Find an expression for [itex]M[/itex].

## Homework Equations

[itex]F = \dfrac{mv^2} r[/itex]

## The Attempt at a Solution

The string will be slanting down slightly to provide a vertical component of tension to keep the mass [itex]m[/itex] in a horizontal circle. Let [itex]\theta[/itex] be the angle that the string makes to the vertical.

Then [itex]T \sin{\theta} = mg[/itex] and [itex]T \cos{\theta} = \dfrac {mv^2} r[/itex]

So [itex]T = \sqrt{(mg)^2 + \left(\dfrac {mv^2} r \right)^2 }[/itex]

We are told the radius is constant, which happens if the bottom mass [itex]M[/itex] is in equilibrium, so [itex]T = Mg[/itex].

So [itex]M = m \sqrt{\left(\dfrac{v^2} {r g}\right)^2 + 1}[/itex]

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