How is angular energy equivalent to linear energy?

Answer 1

There are a number of angular equvalents in physics. Here are some:

Distance, x (meter) is angle, b (radian)
Force, F (Newton) is torque, T (Newton-meter)
Velocity, v(meter/second) is angular velocity, w (radian/second)
Acceleration, a (meter/second^2) is angular acceleration, z (radian/second^2)

So far these are all straight forward.

To make Newton's law (F=m*a) work in angular terms, we need to define an angular mass. It is defined by considering all the differential masses in an object and how they under go angular acceleration when subjected to a torque (an angular force). Without too much trouble, you can show that this angular mass is:

S r^2 dm (S means integration)

This mass property is called moment of inertia and is represented by I. It just takes into account the fact that mass located farther from the axis is harder to accelerate.

With that defined, all the other properties fall into place.

Mass m (kg) becomes moment of inertia I (kg-m^2)
Newton's Law F=ma becomes T=Iz
Kinetic energy E = (m*v^2)/2 becomes E = (I*w^2)/2
Momentum P = m*v, is angular momentum H = I * w

Answer 2

translational kinetic energy = 1/2 (mass) (velocity)^2

(velocity)^2 = (angular speed)^2 / (radius)^2
(mass) = (moment of inertia)^2 x (radius)^2

therefore,
angular kinetic energy = 1/2 (moment of inertia) (angular speed)^2