A conical pendulum is formed by attaching a ball of mass $ m $ to a string of length $ \ell $, then allowing the ball to move in a horizontal circle of radius $ r $. The following figure shows that the string traces out the surface of a cone, hence the name.

Question

A conical pendulum is formed by attaching a ball of mass $ m $ to a string of length $ \ell $, then allowing the ball to move in a horizontal circle of radius $ r $. The following figure shows that the string traces out the surface of a cone, hence the name.(a) Find an expression for the ball&#8217;s angular speed $ \omega $. Express your answer in terms of the variables $ \ell $, $ r $, and the free-fall acceleration $ g $.
, (b) Find an expression for the tension $ T $ in the string. Express your answer in terms of the variables $ \ell $, $ m $, $ r $, and the free-fall acceleration $ g $.
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Jason · Accepted Answer

$\boxed{\omega=\sqrt{\frac{g}{\sqrt{\ell^2-r^2}}}}$
$\boxed{T=\frac{mg\ell}{\sqrt{\ell^2-r^2}}}$.

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