physics question i need help on?

Question

A ring of mass 8 kg and radius 4 cm and cylinder of mass 3 kg and radius 2 cm are placed beside each other at the top of an inclined ramp. They are both released at the same time and roll without slipping down the ramp. Which object (ring or cylinder) reaches the bottom first?


a)they reach the bottom at the same time
b)ring reaches the bottom first
c)cylinder reaches the bottom first
d)insufficient information to determine

Answer 1

Moment of inertia of ring = Ir = Mr*r(rng)^2
Moment of inertia of cyl = Ic =(Mc*r(cyl)^2)/2

Torque on ring = Tr = Mr*r(rng)*g/sinø
Torque on cyl Tc = Mc*r(cyl)*g/sinø

Acceleration (angular) ring = Tr/Ir
=[Mr*r(rng)*g/sinø] / [Mr*r(rng)^2]
=(g*sinø)/r(rng)
Acceleration (angular) cyl = Tc/Ic = 2*(g*sinø)/r(cyl)

Acceleration (linear) of center of mass = accel(ang)*r

a(rng) = g*sinø
a(cyl) = 2*g*sinø

The cylinder has higher acceleration, so reaches bottom first.