# Conservation of Energy Equation

1. Oct 29, 2013

### oneplusone

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).

Last edited: Oct 29, 2013
2. Oct 29, 2013

### mikeph

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.

3. Oct 30, 2013

### stevendaryl

Staff Emeritus
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:

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