Question

A roller is made by joining together two cones at their vertices O. It is kept on two rails AB and CD which are placed asymmetrically (see figure), with its axis perpendicular to CD and its centre O at the centre of the line joining AB and CD (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to

• A. go straight
• B. turn right
• C. turn left
• D. turn left and right alternately

Answer 1

The correct option is C turn left

The image attached here is the front view of the roller. The rails intersect the plane of the paper on both sides where normal reactions are drawn.

We know that the roller does not lift off from the rails.
So, $N_1\cos \theta \times r_1=N_2\cos \theta \times r_2$
$\therefore N_1{r}_1=N_2{r}_2$
As the roller moves forward, $r_1$ becomes smaller.
So, $N_1$ increases. Hence, $N_1>N_2$
Net force in the horizontal direction becomes
$N_1\sin \theta -N_2\sin \theta =(N_1-N_2)\sin \theta$ towards the left.
Since $N_1>N_2$, net force is towards the left and hence the roller moves towards left.