Rotational Motion Question 29
Question: A thick-walled hollow sphere has outside radius $ R _0. $ It rolls down an incline without slipping and its speed at the bottom is $ v _0. $ Now the incline is waxed, so that it is practically frictionless and the sphere is observed to slide down (without any rolling). Its speed at the bottom is observed to be $ 5v _0/4. $ The radius of gyration of the hollow sphere about an axis through its centre is
[JEE ONLINE 26-05-2012$
Options:
A) $ 3R _0/2 $
B) $ 3R _0/4 $
C) $ 9R _0/16 $
D) $ 3R _0 $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] When body rolls dawn on inclined plane with velocity at bottom then body has both rotational and translational kinetic energy Rolling Without Slipping (Rotational + Translational Kinetic Energy): When the hollow sphere rolls down the incline without slipping, both translational and rotational kinetic energy are involved. The potential energy is converted into both linear and rotational kinetic energy. The moment of inertia for a hollow sphere is given by $(I = \frac{2}{5} m R_0^2)$, where (m) is the mass of the sphere and (R_0) is the outside radius. The angular velocity $(\omega_0)$ is related to the linear velocity (v_0) as $ (\omega_0 = \frac{v_0}{R_0})$. Sliding Without Rolling (Only Linear Kinetic Energy): When the incline is waxed and the sphere slides without rolling, only linear kinetic energy is involved. The potential energy is converted entirely into linear kinetic energy. The speed at the bottom of the incline (when sliding without rolling) is given as $(\frac{5v_0}{4})$.
