Answer to Question #138931 in Classical Mechanics for dein

Question

Three objects, a hoop, a disk, and a solid sphere start from the top of a hill and roll without slipping down to the bottom. Which one of the following statements is true?
A. The sphere reaches the bottom first.
B. The disk reaches the bottom first.
C. The object with the greatest mass reaches the bottom first.
D. All three reach the bottom at the same time.
E. The hoop reaches the bottom first.

Answer 1

Thetime of motion

$$t=\sqrt{\frac{2L}{g\sin\theta}\left(1+\frac{I}{mr^2}\right)}$$

The moment of inertia

$I_{hoop}=mr^2$

$I_{disk}=\frac{1}{2}mr^2$

$I_{hoop}=\frac{2}{5}mr^2$

Hence, the time of motion is smallest for the sphere.

Answer: A. The sphere reaches the bottom first.