# Question regarding work-energy theorem

The Work Energy theorem states that work $$W$$ done is equal to the change in kinetic energy $$KE$$.

$$W= Δ{KE}$$

But say some work I do is stored as Potential Energy, in this case, the some work done is used as potential energy and all work isn't converted to kinetic energy. How does this statement remain true in the case of work being stored as Potential Energy?

• You have misstated the theorem. It is the net work that equals the change in kinetic energy. I.e., the work done by the net force on the body. Net force means you have to sum up all the forces acting on the body.
– hft
Oct 20, 2021 at 7:31

Potential energies come from work done by conservative forces. The work-energy theorem includes all work done by all (mechanical) forces, so:

$$\underbrace{W_\text{conservative}+W_\text{other}}_W=\Delta K.$$

Remember that work can be negative, such as when gravity pulls downwards while you lift something up.

Sometimes, instead of referring to the work done by conservative forces we rather want to consider the potential energy that they store due to their conservative nature, and then $$W_\text{conservative}=-\Delta U.$$ Then the energy-work theorem is written as:

$$W_\text{other}=\Delta K+\Delta U,$$

and this is actually the general energy conservation law (for mechanical forces). To avoid confusion remember that potential energy and work by a conservative force are two sides of the same coin - we use the terms more or less interchangable depending on scenario, and you can invoke the work-energy theorem or the general energy conservation law when you feel for it.

When all works of all forces are taken into account, then

sum of all works of all forces acting in the system = change of kinetic energy of the system.

When works of conservative forces (gravity, or spring force) are not taken into account, then

sum of works = change of kinetic energy + change of potential energy.

In the special case the work is done slowly, kinetic energy may not change at all and in that case, work equals change of potential energy. For example, when weight is slowly pulled up in gravity.

Potential energy is used to get work done.

Some forces are called conservative, meaning they have an associated potential energy such that $$W_\text{done by force}=-\Delta U_\text{of the force}$$ where $$W$$ is the work and $$U$$ is the potential energy.

The net work, which is the sum of all the works, gives you the change in kinetic energy.

An easy way to think of it is imagining an object falling under only the influence of gravity (in which case the net work = work by gravity since there is only one force acting on the object).

At a height $$H$$, the object will have some amount of potential energy $$U_i=mgH$$. The object falls, and its potential energy decreases to $$U_f = mgh$$ (where $$H>h\geq0$$). So, the change in potential energy is $$\Delta U = mg( h-H)$$. As the object fell, gravity did $$W=-mg(h-H)$$ joules of work on the object. Of course, as the object falls, it speeds up, so its kinetic energy $$\rm KE$$ increases. By the work kinetic energy theorem, the change in kinetic energy equals the net work done, which in this case is $$\Delta {\rm KE}=-mg(h-H)=mg(H-h)$$. So we know the object's kinetic energy increased by $$mg(H-h)$$.