Rotational Work and Kinetic Energy

Work is defined for translational motion and for rotational motion. Work done by rotational motion is given by

W = F s = F (r$\theta$) = (F r) $\theta$ = $\tau\alpha$.

We can also define the average rotational power in analogy to the linear motion case:

P = W/t = $\tau\theta$/t = $\tau\omega$

The work-energy theorem for rotation tells us

W = $\tau\theta$ = I $\alpha\theta$

$\omega$² - $\omega$_o² =2$\alpha\theta$
or $\alpha\theta$ = ($\omega$²-$\omega$_o²)

We can then rewrite the work as

W = I ($\omega$²- $\omega$_o²)/2 =

I$\omega$² -

I$\omega$_o² = K_rot - K_rot,0 = $\Delta$K_rot

So we can write the rotational kinetic energy as

K_rot =

I$\omega$²

We can now combine this result for the rotational kinetic energy and that previously obtained for the translational kinetic energy and get a form that is generally valid for the cases of translation adn rotation and any combination of the two.

An object moving and rotating has a total kinetic energy given by:

K_tot = K_rot + K_trans =

I_CMw² +

m v_CM²

where I_CM is the moment of inertia about the center of mass, and v_CM is the velocity of the center of mass.

Now, remember that we had said that we can write all moments of inertia in the form

I = c $\cdot$ M $\cdot$ R²

where M is the mass of the object, and R is the radius. c is a constant, a simple number between 0 and 1. For a sphere c = 2/5, for a disk c = 1/2, for a hoop c = 1.

Special case:

Rolling without slipping. In this special case, w and v are connected via v = r w. In this special case, the above formula then reduces to:

K_tot =

(1 + c) m v_CM²(for rolling without slipping)