Question

A small 250g collar C can slide on a semicircular rod which is made to rotate about the vertical AB at a constant rate of 7.5rad/s. Determine the 3 values of theta for which the collar will not slide on the rod, assuming no friction b/w the collar and the rod.

Answer 1

We can assume r= 950 as we have not been provided.

$R=rsin\theta=950sin\theta; v=Rw$

$a_0=\frac{v^2}{R}=\frac{R^2w^2}{R}=rsin\theta w^2$

$\sum{Fy}=ma_y \implies Ncos \theta-mg=0 \implies N=\frac{mg}{cos\theta}$

$\sum{Fx}=ma_x \implies Nsin \theta-ma=0 \implies sin \theta=\frac{ma}{N}$

$rsin\theta cos\theta w^2-gsin\theta=0$

$So, cos \theta = \frac{g}{rw^2} or \space sin\theta=0$

$cos\theta=\frac{9.8}{0.95*7.5^2} \implies \theta=79.42^0$

$sin\theta=0 \implies \theta =0^0 or 180^0$