My name is Alexander FufaeV and here I write about:
Rolling Motion
Important Formula
$$W ~=~ \frac{1}{2} \, \class{brown}{m} \, {\class{blue}{v_{\text c}}}^2 ~+~ \frac{1}{2} \, I_{\text c} \, \class{red}{\omega}^2$$
$$W ~=~ \frac{1}{2} \, \class{brown}{m} \, {\class{blue}{v_{\text c}}}^2 ~+~ \frac{1}{2} \, I_{\text c} \, \class{red}{\omega}^2$$
$$ \class{blue}{v_{\text c}} ~=~ \sqrt{ \frac{ 2W ~-~ I_{\text c} \, \class{red}{\omega}^2 }{ \class{brown}{m} } }$$
$$\class{brown}{m} ~=~ \frac{ 2W ~-~ I_{\text c} \, \class{red}{\omega}^2 }{ {\class{blue}{v_{\text c}}}^2 }$$
$$I_{\text c} ~=~ \frac{ 2W ~-~ \class{brown}{m}\, { \class{blue}{v_{\text c}}}^2 }{ \class{red}{\omega}^2 }$$
$$\class{red}{\omega} ~=~ \sqrt{ \frac{ 2W ~-~ \class{brown}{m}\, { \class{blue}{v_{\text c}}}^2 }{ I_{\text c} } }$$
What do the formula symbols mean?
Total energy
$$ W $$ Unit $$ \mathrm{J} = \mathrm{Nm} = \frac{ \mathrm{kg} \, \mathrm{m^2} }{ \mathrm{s}^2 } $$
Total energy of a rolling body is composed of the rotational energy and the kinetic energy of the translational motion.
Velocity
$$ \class{blue}{v_{\text c}} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Linear velocity of the center of mass of the body. This velocity indicates how fast the body is rolling.
Mass
$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$
Total mass of the rolling body.
Moment of inertia
$$ I_{\text c} $$ Unit $$ \mathrm{kg} \, \mathrm{m}^2 $$
Moment of inertia for an axis of rotation passing through the center of mass of the body.
Angular velocity
$$ \class{red}{\omega} $$ Unit $$ \frac{\mathrm{rad}}{\mathrm s} $$
The angular velocity indicates the angle covered per time, that is how fast the body rotates.
Rolling motion refers to the type of motion that a cylindrical or round body performs on a surface as it moves forward by rolling about its own axis. Unlike sliding motion, in which a body glides over a surface, a rolling body actually rotates about its axis as it moves forward.
$$ \begin{align} W ~=~ \frac{1}{2} \, \class{brown}{m} \, {\class{blue}{v_{\text c}}}^2 ~+~ \frac{1}{2} \, I_{\text c} \, \class{red}{\omega}^2 \end{align} $$