Cylinder and a hoop have the same mass and radius. They are released at the same time and roll down a ramp without slipping.

1) Which one reaches the bottom first?

Cylinder

Hoop

Both reach the bottom at the same time

acceleration of a body rolling down an inclined surface with inclination θ

a = gsinθ / (1 + k² / r²)

Here k is radius of gyration of the body rolling down and r is the radius of it.

For hoop

k = r

a = gsinθ / (1 + r² / r²)

= g / 2 x sinθ

0.5 gsinθ

For cylinder

k² = r² / 2

a = gsinθ / (1 + r² / 2 r²)

= 2 / 3 g sinθ

=.67 g sinθ

So acceleration is more for cylinder

Hence cylinder will reach the bottom first.