# Conservation of Energy Equation

## Main Question or Discussion Point

In an AP Physics C course for mechanics, what other variables are usually added to this equation? :

$$U_g+U_{sp}+K+W_{nc} = U_g+K$$

Also, why is a spring's potential energy only on the left hand side? Would it ever go on the right hand side? (final).

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It entirely depends on what you want to model and what the unexplained terms mean. There is no universal equation of conservation of energy, only the principle of conservation and a manifestation of it as an equation specific to a particular situation.

stevendaryl
Staff Emeritus
In an AP Physics C course for mechanics, what other variables are usually added to this equation? :

$$U_g+U_{sp}+K+W_{nc} = U_g+K$$

Also, why is a spring's potential energy only on the left hand side? Would it ever go on the right hand side? (final).
That equation seems to be for a specific problem. It's not true in general.

If you drop a mass onto a vertical spring, then at the moment right before it hits the spring, its total energy at that moment, $E_0$ will be:

$E_0 = U_{g,0} + K_0$

where $U_{g,0}$ is its gravitational potential energy, and $K_0$ is its kinetic energy, at that moment.

The spring will compress under the impact of the mass, and some of that energy will go into the potential energy of the spring, $U_{sp}$. The gravitational potential energy $U_{g}$ will change, and the kinetic energy $K$ will change. There will also be energy lost due to friction (heating the spring), $W_{nc}$. By conservation of energy, the change in total energy of the mass + spring must all go into the non-conservative work $W_{nc}$. So if we let $E_1$ be the total energy after compressing the spring a little, then

$E_1 + W_{nc} = E_0$

where

$E_1 = K_1 + U_{g,1} + U_{sp, 1}$

where $K_1, U_{g,1}, U_{sp,1}$ are the kinetic energy, gravitational potential energy, and spring potential energy at that moment. Putting it all together:

$K_1 + U_{g,1} + U_{sp,1} + W_{nc} = K_0 + U_{g,0}$