Around And Around

Classical Mechanics Level 3

A particle $P$ is attached to a fixed point $O$ , which is $0.8 \, \text{m}$ above a smooth horizontal plane, by a light inextensible string of length $1.0 \, \text{m}$ .

The particle moves in a circle with angular speed $\omega \, \text{rad s}^{-1}$ . Given that $P$ stays in contact with the plane, find the maximum value of $\omega ^2$ .

Note : Let $g = 9.8 \text{m s}^{-2}$ .

The answer is 12.25.

1 solution

A Former Brilliant Member
Apr 25, 2016

Let G be the point exactly below O.
$l(OG)= 0.8 , l(OP) = 1 \to l(PG) = 6$
Let the angle made by the string with the horizontal be $\theta$ , tension in the string be $T$ , normal reaction from fround be N.
$T\cos(\theta) = ml(PG)\omega^{2}$
$T\sin(\theta) + N = mg$
$\therefore N = mg - ml(PG)\tan(\theta)\omega^{2}$
Since it remains in contact,
$N \ge 0$
$\omega^{2} \le \dfrac{g}{l(PG) \tan(\theta)} = \dfrac{9.8}{0.8} = 12.25$

Around And Around

The answer is 12.25.

1 solution

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