Energy Time Graph

Fig. 2 - Here, there is a block \(A\) that is attached to a spring. The block oscillates between displacements of \(-x\) and \(+x\)

Let's say we have a block \(A\) attached to a spring. What happens if we compress it to position \(+x\) and then let it go? When we compress the block to position \(+x\), it will gain elastic potential energy. When the block returns to equilibrium, it converts its potential energy to kinetic energy. The same case will happen when it stretches to position \(-x\); it converts its kinetic energy to elastic potential energy. While going back to the original state, the potential energy is again converted to kinetic energy. The mechanical energy is conserved if the block is in a closed system, meaning that there is no dissipation or addition of energy. That's why the kinetic and potential energies will convert nonstop, but their total combined value will be constant.

Energy and Displacement Time Graphs

Now, let's do an example of Simple Harmonic Motion (SHM). Below are two graphs that describe what is happening to the block \(A\) (from the image above) as it oscillates back and forth.

Displacement Time Graph

This graph shows the position of block \(A\) as a function of time.

Fig. 3 - This is a Position-Time Graph showing the position change of block \(A\) in time. The graph starts from the \(+x\) position because the block is stretched and let go. The block will keep oscillating between \(+x\) and \(-x\)

The motion plots a sine curve. Notice how it has an amplitude of \(x\) and a period of \(4\,\mathrm{s}\). Remember that the amplitude is the height of the curve from \(0\) and the period is the amount of time it takes the curve to go from one peak to another peak.

Kinetic and Potential Energy vs. Time Graph

This graph is the Energy-Time Graph for block \(A\).

Fig. 4 - This is the Energy-Time Graph of block \(A\). While the green curve shows the elastic potential energy, the red curve shows the kinetic energy

With these two graphs, we can make sense of how the block moves, as well as how the block transfers its energy. Let's take it step by step.

1. After the block is compressed to position \(+x\) and then let go, it first passes from the equilibrium point as indicated in the position-time graph between \(0-1\,\mathrm{sec}\). Here it loses all of its potential energy and converts it to kinetic energy.
2. As it goes back to position \(-x\) between \(1-2\,\mathrm{sec}\), it converts its kinetic energy to potential energy as the spring is stretched.
3. At \(t=2s\), the direction of travel reverses and the block is accelerated back towards the equilibrium point.
4. It passes from the equilibrium position by converting its potential to kinetic energy between the \(2-3\,\mathrm{sec}\).
5. And once again, it converts its kinetic energy to potential energy between \(4-5\,\mathrm{sec}\).

This cycle continues forever, since in SHM the effect of friction is ignored and the mechanical energy is conserved.

The Oscillator Energy Formula

During oscillation, while the mechanical energy is constant, the kinetic and potential energies may vary with time. For example, the variation of a linear oscillator's potential energy depends on how compressed or stretched the spring is, which is \(x(t)\mathrm{.}\)

From principles of simple harmonic motion we can express the displacement at any point in time as \(x(t)=x_\text{m} \cos{(\omega t + \phi )}\), where \(x(t)\) is the function of the displacement with respect to time, \(x_\text{m} \) is the maximum displacement, \(\omega \) is the angular velocity, \(t\) is the time, and \(\phi \) is the phase shift of the cosine function.

$$U(t)=\frac{1}{2}\\ kx^2 = \frac{1}{2}\\k(x_\text{m} \cos{(\omega t + \phi )})^2$$

$$U(t) = \frac{1}{2}\\ k x_\text{m} ^2 \cos^2{(\omega t + \phi)}\mathrm{.}$$

On the other hand, the kinetic energy variation depends on the block's velocity. We know that kinetic energy can be found from \(K=\frac{1}{2}\\ mv^2 \). If \(x(t) = x_\text{m} \cos{(\omega t + \phi )}\), we can derive velocity as \(V(t) = - \omega x_\text{m} \sin{(\omega t + \phi )}\).

We can insert this into the kinetic energy formula,

$$K(t)=\frac{1}{2}\\ m(v(t))^2 = \frac{1}{2}\\ m(-\omega x_\text{m} \sin{(\omega t + \phi )})^2$$

$$K(t) = \frac{1}{2}\\ m \omega ^2 x_\text{m} ^2 \sin^2{(\omega t +\phi )}\mathrm{,}$$

and since \(\omega ^2 = \frac{k}{m}\), we can transform the kinetic energy formula into

$$K(t) = \frac{1}{2}\\ k x_\text{m} ^2 \sin^2{(\omega t + \phi )}\mathrm{.}$$

During the oscillation, the total mechanical energy \(E\) is conserved. So, the sum of kinetic energy and potential energy is equal to mechanical energy:

$$E=U+K$$

$$E=\frac{1}{2}\\ k x_\text{m} ^2 \cos^2{(\omega t + \phi )} + \frac{1}{2}\\ k x_\text{m} ^2 \sin^2{(\omega t +\phi )}\mathrm{.}$$

Since \(\frac{1}{2}\\ k x_\text{m} ^2 \) is the common part of the equation, we can rewrite the sum as

$$E=\frac{1}{2}\\ k x_\text{m} ^2 (\cos^2{(\omega t + \phi )} + \sin^2{(\omega t + \phi )})\mathrm{.}$$

It is important to know that \(\sin^2{a} + \cos^2{a} = 1\). So, the sum will be equal to \(E=\frac{1}{2}\\k x_\text{m} ^2 \).

Capacitor Energy-Time Graph

So far, we've really only shown how Energy vs. Time Graphs can help us to understand oscillators. But that is not their only use! Energy-Time Graphs can also aid us in modeling the energy in capacitors.

Calculating Energy Given by Power-Time Graph

There is no real application for the area under an Energy-Time Graph unless we're talking quantum physics and phase shifts. My guess is that the only knowledge you have about quantum physics derives from Avengers: Endgame (which is all wrong: shocker!), so we won't go down that rabbit hole.

The slope of an Energy-Time Graph does tell us something useful, however. But, before we get into that, we need a little background.

Power equals the rate of energy over time. Therefore, power gives us a quantity for how much punch our energy packs. A little bit of energy over a large amount of time will give us little punch. Whereas a large amount of energy over a little amount of time gives us massive punch. A relevant equation for this principle is

$$P=\frac{\Delta E}{\Delta t}\\\mathrm{.} $$

From this knowledge, we glean that the slope of an Energy-Time Graph equals the power! Remember that slope is the rate of change of \(y\) over the rate of change of \(x\). Since the energy is the \(y\) and the time is the \(x\), plugging these into the slope equation yields

$$m=\frac{\Delta E}{\Delta t}\\$$

where \(m\) is the slope. That is the exact equation for power. To illustrate this principle, we'll go over the Energy-Time Graph below.

Fig. 6 - The slope of an Energy-Time Curve is power

Fig. 6 shows that the slope of an Energy vs. Time Graph equals power. To find the power of the above function of energy with respect to time, we need only find the slope:

$$\begin{align*} m&=\frac{\Delta y}{\Delta x} \\ m&=\frac{100\,\mathrm{J}-0\,\mathrm{J}}{1\,\mathrm{s}-0\,\mathrm{s}} = 100\,\mathrm{\frac{J}{s}} \\ m&=\text{Power}=100\,\mathrm{\frac{J}{s}.} \\ \end{align*}$$

Kinetic Energy vs. Time Graph and Simple Harmonic Motion

Now, it's time for some examples.

That was quite the rollercoaster ride! There was a lot of back and forth, some ups and downs, and even more stops and drops. Hopefully, by now, you aren't nauseous from all the physics frenzy, and you're ready to absorb the essential key takeaways from our article on Energy-Time Graphs.