System of Particles and Rotational Motion

After the collision the disk starts to rotate around point C in the XY plane. The maximum change in the height (in m) of its center O is:

Amount of energy loss (in J) in the collision is:

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 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 . If the angular speed at which the large disc rotates is /n, then the value of n is . [figure]

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=L/n from the mid-point 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 . [figure]

If the collision occurs at time t0 = 0, the value of vcm/(a ) will be .