Question

A disk with a c value of 1/2, a mass of 4 kg, and radius of 0.28 meters, rolls without slipping down an incline with has a length of 7 meters and angle of 30 degrees. At the top of the incline the disk is spinning at 24 rad/s. How fast is the disk spinning (the center of mass) at the bottom of the incline in rad/s?

Answer 1

here,

the mass of disk , m = 4 kg

radius , r = 0.28 m

length , l = 7 m

theta = 30 degree

initial angular speed , w0 = 24 rad/s

let the final angular speed be w

using conservation of Mechanical energy

PEi + KEi = PEf + KEf

m * g * l * sin(theta) + (0.5 * I * w0^2 + 0.5 * m* u^2) = (0.5 * I * w^2 + 0.5 * m* v^2)

m * g * l * sin(theta) + (0.5 * (0.5 * m * r^2) * w0^2 + 0.5 * m* u^2) = (0.5 * (0.5 * m * r^2) * w^2 + 0.5 * m * (r * w)^2)

g * l * sin(theta) + 0.75 * r^2 * w0^2 = 0.75 * r^2 * w^2

9.81 * 7 * sin(30) + 0.75 * 0.28^2 * 24^2 = 0.75 * 0.28^2 * w^2

solving for w

w = 34.06 rad/s

the final angular speed is 34.06 rad/s