3 questions from 2022
A solid sphere of mass \( 1 \, \text{kg} \) and radius \( 1 \, \text{m} \) rolls without slipping on a fixed inclined plane with an angle of inclination \( \theta = 30^\circ \) from the horizontal. Two forces of magnitude \( 1 \, \text{N} \) each parallel to the incline, act on the sphere, both at distance \( r = 0.5 \, \text{m} \) from the centre of the sphere, as shown in the figure. The acceleration of the sphere down the plane is ______ \( \text{m s}^{-2} \). (Take \( g = 10 \, \text{m s}^{-2} \).) 
At time \( t = 0 \), a disk of radius \( 1 \, \text{m} \) starts to roll without slipping on a horizontal plane with an angular acceleration of \( \alpha = \dfrac{2}{3} \, \text{rad s}^{-2} \). A small stone is stuck to the disk. At \( t = 0 \), it is at the contact point of the disk and the plane. Later, at time \( t = \sqrt{\pi} \, \text{s} \), the stone detaches itself and flies off tangentially from the disk. The maximum height (in \( \text{m} \)) reached by the stone measured from the plane is \( \dfrac{1}{2} + \dfrac{x}{10} \). The value of \( x \) is ________. [Take \( g = 10 \, \text{m s}^{-2} \)]
A flat surface of a thin uniform disk \( A \) of radius \( R \) is glued to a horizontal table. Another thin uniform disk \( B \) of mass \( M \) and with the same radius \( R \) rolls without slipping on the circumference of \( A \), as shown in the figure. A flat surface of \( B \) also lies on the plane of the table. The center of mass of \( B \) has fixed angular speed \( \omega \) about the vertical axis passing through the center of \( A \). The angular momentum of \( B \) is \( n M \omega R^2 \) with respect to the center of \( A \). Which of the following is the value of \( n \)? 
