Question

Two particles 1 and 2 are allowed to descend on two frictionless chords $OA$ and $OB$ of a vertical circle, at the same instant from point $O$. The ratio of the velocities of 1 and 2 respectively, when they reach the circumference will be ($OB$ is the diameter)

• A. $\sin \alpha$
• B. $\tan \alpha$
• C. $\cos \alpha$
• D. none of these

Answer 1

The correct option is C $\cos \alpha$

Let $d$ be the diameter

Distance travelled by particle 1, $OA=d\cos \alpha$
Acceleration of particle 1, $a_{OA}=g\cos \alpha$
Along $OA$, velocity of particle 1 using 3rd equation of motion,
$\begin{array}{}{v}_1^2=2\left(g\cos \alpha \right).\left(d\cos \alpha \right)& \text{(1)}\end{array}$
Similarly for particle B,
$\begin{array}{}{v}_2^2=2gd& \text{(2)}\end{array}$
Dividing equation $\left(1\right)$ and $\left(2\right)$, we get
$\frac{v_1}{v_2}=\cos \alpha$