2 questions from 2024
A disc of mass \( M \) and radius \( R \) is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass \( M \) and radius \( \frac{R}{2} \) is fixed to the motor’s thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed \( \omega \). If the angular speed at which the large disc rotates is \( \frac{\omega}{n} \), then the value of \( n \) is __________. 
A thin uniform rod of length \( L \) and certain mass is kept on a frictionless horizontal table with a massless string of length \( L \) fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point \( O \). If a horizontal impulse \( P \) is imparted to the rod at a distance \( x=\frac{L}{n} \) from the midpoint of the rod (see figure), then the rod and string revolve together around the point \( O \), with the rod remaining aligned with the string. In such a case, the value of \( n \) is ________.
