Question

A hoop and a disk with uniform mass distribution have the same radius but the total masses are not known. Can they both roll down a ramp without slipping and reach the bottom in the same time? And if so, what can you deduce about the relative masses?
a) The hoop and disk have the same mass.
b) The hoop is heavier, twice the mass of the disk.
c) The disk is heavier, twice the mass of the hoop.
d) They cannot reach the bottom at the same time regardless of mass.
e) The disk is lighter, three-fourth the mass of the hoop

Answer 1

Answer:

Explanation:

radius of hoop and the radius of disk is same = R

Let the mass of hoop is M and the mass of disk is M'.

As they reach the bottom of teh surface in same time so they travel equal distance thus, they have same acceleration.

The acceleration is given by

$a=\frac{gSin\theta }{1+\frac{I}{MR^{2}}}$

As the acceleration is same so that the moment of inertia is also same.

Moment of inertia of disk = moment of inertia of hoop

1/2 x mass of disk x R² =  mass of hoop x R²

So, mass of disk = 2 x mass of hoop

Option (c) is correct.