A 310N sphere 020 m in radius rolls without slipping 60 m do

A 310-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 34° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?

Solution

by conservation of energy

initial energy = final energy

mgh = 0.5 * I * w^2 + 0.5 * mv^2

mg * 6 * sin(34) = 0.5 * (2/5) * m * r^2 * v^2 / r^2 + 0.5 * mv^2

g * 6 * sin(34) = 0.5 * (2/5) * v^2 + 0.5 * v^2

9.8 * 6 * sin(34) = 0.5 * (2/5) * v^2 + 0.5 * v^2

v = 6.853 m/s

velocity = angular velocity * radius

6.853 = angular velocity * 0.2

angular velocity = 34.265 rad/sec