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

Answer:

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 $I=\frac{MR^2}{2}$

Kinetic Energy of cylinder $K.E.=\frac{1}{2}Mv^2$

Rotational energy $R.E.=\frac{1}{2}I\omega ^2$

for rolling

$v=\omega \times r$

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

$R.E.=\frac{1}{2}\times \frac{MR^2}{2}\times (\frac{v}{R})^2=\frac{1}{4}Mv^2$

Total kinetic Energy $=\frac{1}{2}Mv^2+\frac{1}{4}Mv^2=\frac{3}{4}Mv^2$