Physics in Motion

Rotational Kinetic Energy

When we consider a rotating object, we are able to calculate the total kinetic energy associated with the rotational mass of all of the particles within the rotating object:

$$ K_R = \sum K_i = \sum \frac{1}{2} m_i v^{2}_{i} = \frac{1}{2} \sum m_i r^{2}_{i} \omega^{2} $$

$$ = \frac{1}{2} \left( \sum m_i r^{2}_{i} \right) \omega^2 $$

where the variables in the brackets are called the moment of inertia:

$$ I = \sum m_i r^{2}_{i} $$

This can be though of as rotational mass, since it palys the same role for rotation that mass does in non-rotational motion.

In terms of moment of intertia, the rotational kinetic energy can be expressed:

$$ K_R = \frac{1}{2} I \omega^2 $$

which is analogous to translational kinetic energy.

One important observation is that the moment of intertia depends both on the amount of mass and on how far from the axis the mass is located.

Common equations for various common moments of intertia can be found follow.

Hoop or cylindrical sheel around axis:

$$ M R^2 $$

Solid cylinder or disk:

$$ \frac{1}{2} M R^2 $$

Rod about perpendicular axis through centre:

$$ \frac{1}{12} M R^2 $$

Rod about perpendicular axis through end:

$$ \frac{1}{3} M R^2 $$

Rectangular plate \( a \times b \) about perpendicular axis through center:

$$ \frac{1}{2} M ( a^2 + b^2 ) $$

Solid sphere:

$$ \frac{2}{5} M R^2 $$

Spherical shell:

$$ \frac{2}{3} M R^2 $$

The total kinetic energy of a rolling object is equal to the kinetic energy of translation of the center of mass, plus the kinetic energy of rotation about the center of mass:

$$ KE = \frac{1}{2} I_{CM} \omega^2 +\frac{1}{2} M v^2 $$