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## Homework Statement

A particle,A, of mass,m, hangs by a light inextensible string of length,a from a fixed point O. The string is initially vertical and the particle is then given a horizontal velocity,[itex]\sqrt{nga}[/itex]. Show that it will move round a complete vertical circle in a vertical plane provided [itex]n \geq 5[/itex]

## Homework Equations

Centripetal force=[itex]\frac{mv^2}{r}[/itex]

## The Attempt at a Solution

Well the resultant force of the tension in the string and the component of the weight provides the centripetal force.

[itex]F_c=T-W_{component}[/itex]

[tex]\frac{mv^2}{a}=T-mgcos\alpha...(*)[/tex]

If initially it is vertical then [itex]\alpha=0[/itex] (Doesn't really seem to help)

For the object to make a complete circle, then the string must be taut at the highest point (i.e. when [itex]\alpha=180,T\geq 0[/itex]

From (*)

[tex]T=\frac{mv^2}{a}+mgcos180 \Rightarrow T=\frac{m(\sqrt{nga})^2}{a}-mg[/tex]

So that

T=mgn-mg

For [itex]T \geq 0[/itex] then [itex]mng \geq mg \Rightarrow n \geq 1[/itex]

Which is not what I want to show.