Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time.

Torques of equal magnitude are applied to a hollow cylinder and a soli

1.	Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time.
Answer» Solution :Suppose `I_(1)andI_(2)` be the respective moment of inertia of hollow CYLINDER and the solid sphere and `omega_(1)andomega_(2)` be the respective ANGULAR VELOCITY and `alpha_(1)andalpha_(2)` respective angular accelerations. As can equal torque is applied to both the bodies `I_(1)alpha_(1)=I_(2)alpha_(2)` `:. (I_(1))/(I_(2))=(alpha_(2))/(alpha_(1))` but `(MR^(2))/((2)/(5)MR^(2))=(alpha_(2))/(alpha_(1))` `:.(5)/(2)=(alpha_(2))/(alpha_(1)).....(1)` Now in `omega=omega_(0)+alphat,omega_(0)=0` `omega_(1)=alpha_(1)tandomega_(2)=alpha_(2)t` `:. (omega_(1))/(omega_(2))=(alpha_(1))/(alpha_(2)).....(2)` `:. (omega_(1))/(omega_(2))=(2)/(5)` [From EQN. (1)] `:. omega_(2)gtomega_(1)` `:.` The angular speed of the solid sphere will be greater than that of hollow cylinder.