Question

A disk with a c value of 1/2, a mass of 4 kg, and radius of 0.49 meters, rolls without slipping down an incline with has a length of 9 meters and angle of 30 degrees. At the top of the incline the disk is spinning at 33 rad/s. How fast is the disk spinning (the center of mass) at the bottom of the incline in rad/s?

Answer 1

By energy conservation ,

Kinetic Energy + Potential energy at the top = Kinetic energy at the bottom

0.5iw1^2 + 0.5mv1^2 + mgh = 0.5iw2^2 + 0.5mv^2

0.5*(c*mR^2)w1^2 + 0.5mv1^2 + mgL sin 30 degree = 0.5*(c*mR^2)w2^2 + 0.5mv2^2

0.5*(0.5* mv1^2) + 0.5* mv1^2 + mgL/2 = 0.5*0.5* mv2^2+0.5* mv2^2 as v=wR for pure rolling

0.75mv1^2 + mgL/2 = 0.75* mv2^2

    0.75*v1^2 + gL/2 = 0.75*v2^2

0.75*(0.49*33)^2 + 9.8*9/2 = 0.75*(0.49*w2)^2

w2 = 36.52 rad/s answer