Energy
Our operational definition of energy was:
Something has energy ...
-
...if it has the potential to physically change itself or its environment.
Gravitational energy?
Does a book on a stack high above the floor have the "potential to physically change itself or its environment"?
If so, what variables (characteristics of an object) does that depend on?
Dropping boulders
What combination of weight, $w$, and height, $h$, above the road expresses the destructive potential (we'll call this the "Gravitational Energy"=GravE)) of a boulder?
- $\frac wh$
- $\frac hw$
- $wh$
- $\frac{1}{wh}$
- $h + w$
To decide among these formula..., Ask each other first: Which do I think has more destructive potential?:
- A book weighing two pounds that is 2 feet above the floor?
- A book weighing one pound that is one foot above the floor?
Try "plugging" these values for $w$ and $h$ into the different formulas, and compare the results for the two books. show / hide
We considered that something that "weighs" more (say, 200 lbs, instead of 20 lbs) has more destructive potential than a lighter object.
We considered that something that falls from a greater height (say 3 meters instead of 2 meters) has more destructive potential than something that falls only a short distance.
So we want the mathematical combination of $w$ and $h$ that gives us a large number when $w=2$ and $h=3$, and a small number when $w=1$ and $h=1$. Let's test and compare...
| test... | with $w=1$, $h=1$ | with $w=2$, $h=3$ |
|---|---|---|
| $\frac wh$ | $\frac{1}{1}=1$ | $\frac{2}{3}=0.67$ |
| $\frac hw$ | $\frac{1}{1}=1$ | $\frac{3}{2}=1.5$ |
| $wh$ | 1*1=1 | 2*3=6.0 |
| $\frac{1}{wh}$ | $\frac{1}{1*1}=1$ | $\frac{1}{2*3}=\frac 16=0.17$ |
Here is a picture of a very large "battery" for storing GravE:
Is there a battery that could store enough energy for a city?
Weight
What do we mean by "weight"?
Two aspects:
- How much *stuff*...that is "atoms" (and what kind).
- What planet are we on? (table)
An astronaut wearing a spacesuit and a backpack with life support equipment is weighed on scale on Earth, and comes out at 300 lbs. But On the moon he weighs just 50 lbs.
--OR-- even on Earth there are variations in what a scale would read:
Are we floating in the ocean? or standing on land?
Weight = mass * strength of gravity ($g$) = $mg$.
Your mass does not change from planet to planet, but the strength of gravity may change.
With your mass in kg and the strength of gravity in m/sec^2 (the table) this expression gives the force of gravity acting on you in (metric units) "Newtons" or (english units) "pounds".
So our final expression for gravitational energy, which also takes into account your planet is
$$\text{GravE} = m g h$$
Units:
- If $m$ is in kilograms,
- $g$ is in m/sec/sec (9.8 m/sec/sec on Earth),
- $h$ is in meters
$\Rightarrow$ energy will be in units of $\frac{\text{m}^2\cdot \text{kg}}{\text{set}^2}=$ Joule.
Practice problem
How much energy does a ceramic mug on the top floor of the Ad Building have (in Joules)?
- Estimate the mass of a mug in kilograms
- Estimate the height of a windown on the top (3rd) floor of the Ad building.
- Use $g=9.8$ m/sec^2, and calculate: $$\text{GravE} = mgh.$$
There is a sense in which gravitational energy is a relative quantity:
- If I drop the mug from the top floor of the Ad building out the window, I might want to know how much energy it has, relative to the ground.
-OR- - Maybe the mug is indoors, and I drop it onto my foot on the floor of a room on the top floor.