# Unit 7 - Energy Storage

This unit is an exploration of how the energy concept is used in physics. This page explains how we derived the models for energy storage.

CONCEPTS FOR UNIT 7: ENERGY

**Paradigm**: We graphed Force vs. stretch for a coil spring. the result looked something like this:

What does this have to do with energy? As the force increased, the stretch increased, storing more Elastic Energy (E_{el}) in the spring. This is evidenced by the fact that the spring snaps back harder the more it is stretched, and has the capability to make more change happen the more it is stretched.

From this graph, we derived Hooke's Law: F=kx. Or Force (exerted by the spring)=k (spring constant) times stretch.

The slope is "k," the spring constant. This is the force required to stretch (or compress) the spring one meter. A stiffer spring has a higher spring constant.

Hooke's law applies to springs with equally spaced coils, when the coils are not touching each other.

**Elastic Energy**: Energy stored in "springy things"

The energy stored in the spring increased as the spring stretched. This is shown by the area of the graph increasing as we go from any x_{1} to any x_{2}. How do we calculate the area of the graph?

Area of a triangle is 1/2base*height. So, in going from x_{1} to x_{2}, the change in Elastic Energy is the change in area, or 1/2F*x_{2}-1/2F*x_{1}. But, F=kx, so this becomes:

delta Elastic Energy = 1/2kx_{2}^{2 }- 1/2kx_{1}^{2}

Both the area of the graph and energy have units of N*m, or Joules. Note how even though each graph shows a change in stretch of 5 m, it takes 3 times as much energy to stretch the spring from 5 to 10 m as it does from 0 to 5 m.

**Gravitational Energy**: Energy stored in the interaction of masses

Let's examine gravitational energy. Since we assume the force of gravity is constant near the surface of the earth (and it pretty nearly is constant), could make a graph of gravitational force vs. height for an object like the one below:

This graph shows that the gravitational force is constant as we increase the height of the object. So what's the point of this? Well, just like before we have a graph of Force vs. Position. And once again the area is the energy stored in the earth-object system. So, since the area bounded by the "curve" is a rectangle, the energy is the area of that rectangle, or L x W, or F_{grav} x delta h.

Since, by the gravitational force rule, the F_{grav} = mg, then we come up with this definition:

E_{grav} = m*g*deltah=mg(h_{2}-h_{1})

**Kinetic Energy**: Energy stored in moving objects

We investigated the energy stored in moving objects, or Kinetic Energy (E_{K}) using our results for Elastic Energy.

First, we determined the spring constant of an elastic cord that we had in the lab, using the method from the paradigm lab.

We attached the elastic cord to a lab cart. Then we stored energy in the system by stretching the cord. We released the cart, passing it through a photogate at them moment it reached its peak speed. We were able to collect data for velocity and stretch. Using the spring constant from before, we calculated the elastic energy stored in the cord for each stretch. We assumed that all of that energy was stored in the kinetic energy of the lab cart at the moment the cord went slack:

initial E_{el}= final E_{k}

Note that the graph looks like a parabola. Next step, convert it into a linear graph because they are the easiest to understand. We squared speed to do this:

The slope of the graph was very nearly one half of the mass of the lab cart, leading us to decide that the equation for kinetic energy is:

E_{k}=1/2*mass*deltavelocity^{2} or E_{k}=1/2mv_{2}^{2}-1/2mv_{1}^{2}

**Working**: Transferring energy in or out of a system

Let's keep it simple. Consider the system to be a book. The book is at rest on a table. You move the book across the table at constant speed by tugging gently on a string attached to the book. So, initially you store some kinetic energy in the book, but that doesn't change as you drag the book across the table. However, you must keep using chemical energy stored in your body to keep the book moving. You are working on the system (the book, remember). The energy transferred is the area of the F vs. delta x graph. In this case, we considered a constant force graph, but the energy transferred is still the area, even if it's a varying force. "Of course," you say. "I knew that." That's why I like you so much.

**Random Notes**

What is energy? As a simple working definition, we can say simply that it is a substance-like quantity that allows change to happen.

Why substance-like? Well, like matter, energy is conserved. Meaning you can put it in different places, but you can't make it disappear.

You can think of energy like money. You can keep it in the bank, under your mattress, or in your wallet, but it is still money.

Why do we care about energy? Energy is the account book for change in the universe.

Website bonus. The first person from each class to mention they read this page while we are studying Unit 7 will receive bonus points.

16-Mar-2006