Angular Work

For reference, work is the energy that is given to an object when you apply a force along a given direction of motion.

So, how does this translate into a rotational equivalent? It's actually a lot easier than you think.

Remember that work is just force acting along a direction of motion, as shown below:

Let's apply this principle to a rotating system. We know the above is true for systems undergoing strictly linear motion but we can use some algebraic manipulation to make it applicable to a rotating system.

It may seem like this new equation is very different from the one above. However, it actually isn't.

We simply took the equation for linear work and multiplied and divided the radius of the system to it. If we multiply an equation by a quantity but also divide it by the same quantity, then the equation is unchanged, making this algebraically consistent.

If we didn't technically change the equation, then what did we do? Well, the quantities inside both parentheses can actually be substituted into two very relevant quantities.

It may seem like this new equation is very different from the one above. However, it actually isn't.

We simply took the equation for linear work and multiplied and divided the radius of the system to it. If we multiply an equation by a quantity but also divide it by the same quantity, then the equation is unchanged, making this algebraically consistent.

If we didn't technically change the equation, then what did we do? Well, the quantities inside both parentheses can actually be substituted into two very relevant quantities.

We can substitute the two equations given above into the net work equation to get the equation below:

This equation is the net work done by a net torque acting along a given range of angular displacement. This is analogous to the definition of linear work.

You might be able to intuitively get this equation because torque is the rotational equivalent to force and angular displacement is the rotational equivalent to linear displacement. However, the derivation (hopefully) makes it clearer as to why the equation is exactly the way it is.

Above is an alternate definition of net work done by a torque by applying Newton's Second Law(rotational) and substituting accordingly. An image of how net rotational work works(pun intended) is given below:

To get the instantaneous kinetic energy of an object, you just remove the delta from the right side. This makes sense as the moment of inertia is multiplied by angular velocity squared similar to how mass is multiplied by linear velocity squared for linear kinetic energy.