Question

A particle is to slide along the horizontal circle on the inside of a conical funnel as shown. For the execution of this motion the velocity of the particle in terms of $r$ and $\theta$ in the Fig. is

• A. $\sqrt{gr\tan \theta }$
• B. $\sqrt{gr\cot \theta }$
• C. $\sqrt{gr\sin \theta }$
• D. $\sqrt{gr\cos \theta }$

Answer 1

The correct option is B $\sqrt{gr\cot \theta }$

Equation of motion:

$N\sin \theta =mg$

$⟹N=\frac{mg}{\sin \theta }$

and $N\cos \theta =\frac{m{V}^2}{r}$

$⟹V^2=\frac{r}{m}\cos \theta \times \frac{mg}{\sin \theta }$

$⟹V^2=rg\frac{\cos \theta }{\sin \theta }$

$⟹V^2=rg\cot \theta$

$⟹V=\sqrt{rg\cot \theta }$