Using the Law of Conservation of Energy

Using the Law of Conservation of Energy

Mechanical energy is conserved (in the absence of friction). Therefore we can say that the sum of the \({E}_{P}\) and the \({E}_{K}\) anywhere during the motion must be equal to the sum of the the \({E}_{P}\) and the \({E}_{K}\) anywhere else in the motion.

We can now apply this to the example of the suitcase on the cupboard. Consider the mechanical energy of the suitcase at the top and at the bottom. We can say:

\begin{align*} {E}_{M1} & = {E}_{M2} \\ {E}_{P1} + {E}_{K1} & = {E}_{P2} + {E}_{K2} \\ mgh + \frac{1}{2}m{v}^{2} & = mgh + \frac{1}{2}m{v}^{2} \\ \left(\text{1}\text{ kg}\right)\left(\text{9.8}\text{ m·s$^{-2}$}\right)\left(\text{2}\text{ m}\right) + 0 & = 0 + \frac{1}{2}\left(\text{1}\text{ kg}\right)\left({v}^{2}\right) \\ \text{19.6} & = \frac{1}{2}\left({v}^{2}\right) \\ {v}^{2} & = \text{39.2}\text{ m$^{2}$·s$^{-2}$} \\ v & = \text{6.26}\text{ m·s$^{-1}$} \end{align*}

The suitcase will strike the ground with a velocity of \(\text{6.26}\) \(\text{m·s$^{-1}$}\).

From this we see that when an object is lifted, like the suitcase in our example, it gains potential energy. As it falls back to the ground, it will lose this potential energy, but gain kinetic energy. We know that energy cannot be created or destroyed, but only changed from one form into another. In our example, the potential energy that the suitcase loses is changed to kinetic energy.

The suitcase will have maximum potential energy at the top of the cupboard and maximum kinetic energy at the bottom of the cupboard. Halfway down it will have half kinetic energy and half potential energy. As it moves down, the potential energy will be converted (changed) into kinetic energy until all the potential energy is gone and only kinetic energy is left. The \(\text{19.6}\) \(\text{J}\) of potential energy at the top will become \(\text{19.6}\) \(\text{J}\) of kinetic energy at the bottom.

The activity with the marble rolling down the pipe shows very nicely the conversion between gravitational potential energy and kinetic energy. In the first instance, the pipe was held relatively low and therefore the gravitational potential energy was also relatively low. The kinetic energy at this point was zero since the marble wasn't moving yet. When the marble rolled out of the other end of the pipe, it was moving relatively slowly, and therefore its kinetic energy was also relatively low. At this point its gravitational potential energy was zero since it was at zero height above the table top.

In the second instance, the marble started off higher up and therefore its gravitational potential energy was higher. By the time it got to the bottom of the pipe, its gravitational potential energy was zero (zero height above the table) but its kinetic energy was high since it was moving much faster than the first time. Therefore, the gravitational potential energy was converted completely to kinetic energy (if we ignore friction with the pipe).

In the case of the pipe being held higher, the gravitational potential energy at the start was higher, and the kinetic energy (and velocity) of the marble was higher at the end. In other words, the total mechanical energy was higher and and only depended on the height you held the pipe above the table top and not on the distance the marble had to travel through the pipe.