Question

"A uniform cylinder of mass M and radius R is rolling without slipping. The velocity of its center of mass is v. What is the cylinder's total kinetic energy?"

Answer 1

Step-by-step explanation:

Given

mass of cylinder is M

radius R

velocity of center of mass is v

As there is no slipping therefore cylinder will rotate as well as translate

Moment of inertia of cylinder
`I=(MR^2)/(2)`

Kinetic Energy of cylinder
`K.E.=(1)/(2)Mv^2`

Rotational energy
`R.E.=(1)/(2)I\omega ^2`

for rolling

`v=\omega * r`

where
`\omega =angular\ velocity\ of\ cylinder`

`R.E.=(1)/(2)* (MR^2)/(2)* ((v)/(R))^2=(1)/(4)Mv^2`

Total kinetic Energy
`=(1)/(2)Mv^2+(1)/(4)Mv^2=(3)/(4)Mv^2`