Question

Figure shows a smooth fixed track with two straight parts connected by a semicircular arc ABC. The shown track is placed in a vertical plane. A small ball is given an initial speed 'u' along the track as shown. The ball loses contact with the track somewhere in the part BC and collides with the track at A during subsequent motion. Now answer the following questions.

Find the value of 'u'.

Answer 1

This ques had been asked in dps school, rithala, delhi in their annual exam of physics

Answer 2

The value of u is (C) $\sqrt{\frac{7}{2}Rg }$

From the derivations of the motion of the ball in a circular rod, we get the following values,

• u$_{min$ at the lowest point of the circular track = $\sqrt{Rg}$ [where R is the radius of the circular rod/ track]
• u$_{min$ at the middle point of the circular track (where the angle is 90°)

       = $\sqrt{3Rg}$

• u$_{min$ at the highest point of the circular track = $\sqrt{4Rg}$
• u$_{min$ at the middle point of the circular track (where the angle is 180°)

       = $\sqrt{2Rg}$

Given, the ball loses contact with the track somewhere in the part BC and collides with the track.

The point lies in between the speed range of  $\sqrt{3Rg}$ and $\sqrt{4Rg}$.

Now, do the back calculation.

According to the options, only one option satisfies this condition, i.e.,    $\sqrt{\frac{7}{2}Rg }$

So, the 'u' will be  $\sqrt{\frac{7}{2}Rg }$.