Two disks A and B with identical mass (m) and radius (R) are initially at rest. They roll down from the top of identical inclined planes without slipping. Disk A has all of its mass concentrated at the rim, while Disk B has its mass uniformly distributed. At the bottom of the plane, the ratio of velocity of the center of disk A to the velocity of the center of disk B is.

The rod PQ of length L = 2 m, and uniformly distributed mass of M = 10 kg, is released from rest at the position shown in the figure. The ends slide along the frictionless faces OP and OQ. Assume acceleration due to gravity, g = 10 m/s² . The mass moment of inertia of the rod about its centre of mass and an axis perpendicular to the plane of the figure is (ML² /12). At this instant, the magnitude of angular acceleration (in radian/s² ) of the rod is ____________.

A system of particles in motion has mass center $$G$$ as shown in the figure. The particle $$i$$ has mass $${m_i}$$ and its position with respect to a fixed point $$O$$ is given by the position vector $${r_i}$$ . The position of the particle with respect to $$G$$ is given by the vector $${\rho _i}$$ . The time rate of change of the angular momentum of the system of particles about $$G$$ is (The quantity $$\mathop {{\rho _i}}\limits^{..} $$ indicates second derivative of $${\rho _i}$$ with respect to time and likewise for $${r _i}$$).

An inextensible massless string goes over a frictionless pulley. Two weights of 100 N and 200 N are attached to the two ends of the string. The weights are released from rest, and start moving due to gravity. The tension in the string (in N) is __________ .