K= p2(p square) / 2m. What is the analogous relation between rotational kinetic energy and angular momentum?

Answer 1

L = angular momentum

I = moment of inertia

ω = angular velocity

Then since:
L = I*ω
and:
KE = I*ω^2
then:
KE = L^2/(2*I)

But this is more complicated when you have a complex-shaped solid: In which case I is a tensor, so you can't divide by it. Instead, I think the solution is:
KE = ω*I*ω/2

Answer 2

If p = mv; then you have K = m^2 v^2/2 m = 1/2 mv^2, which is linear kinetic energy for a point mass m traveling v velocity. (you really must define your terms when asking a question)

Rotational kinetic energy E = 1/2 I w^2; where I is the angular moment of inertia and w is anglular velocity of a body of point mass m rotating around an axis a distance r away. I alpha = Fr; so that I = Fr/alpha; where alpha is the angular acceleration, F is the force acting on the point mass perpendicular to and at a distance r from the axis of spin.

Angular momentum L = pr = mvr; where p is the linear momentum acting on a point mass m perpendicular to and at a distance r from the spin axis. And the change in L over time is called torque (T = dL/dt). Therefore, we have T = dL/dt = dp/dt r = mr dv/dt = ma r = Fr; where F is just the force acting perpendicular to and on the distance r from the spin axis for the point mass m.

[NB: L is a pseudo vector found by the so-called right hand rule for cross multiplying. If this is meaningless, don't worry about it; you'll get this later if you stay with physics.]

Thus we have I = Fr/alpha = ma r/alpha and a = alpha r; so that I = m alpha r^2/alpha = m r^2 which is the equation most of us have memorized for the angluar moment of inertia for a point mass rotating around r.

Then the total kinetic energy of that mass m is T = E + K, which is the sum of the linear and rotational kinetic energies. This can apply quite nicely to an electron of point mass m rotating around an atom with w angular velocity so it has E energy while the whole atom is moving along with velocity v; so that the electron also has linear K.

Bottom line, the kinetic energy equations are similar. But m becomes I and v becomes w in the angular version. And the real physical difference is that the angular version accounts for the rotation radius r, while the linear version does not.