Power

In which we find out the difference between ...

power vs. energy.

Gravitational energy comparison

Gravitational energy: $$\text{GravE}=wh=mgh.$$

  • $m$ is your *mass* in kilograms. (2.2 kilograms / pound)
  • $g$ is a constant, characteristic of the planet you're on. $$\text{Earth: } g=9.8 \text{ meters/sec^2}.$$
  • $h$ is the height in meters, or the height difference between two places on Earth

So, the units of energy are... $$\text{Energy, }m*g*h = [\text{kg}][\text{m}/\text{sec}^2][\text{m}]=\left[\frac{\text{kg}\cdot\text{m}^2}{\text{sec}^2}\right]\equiv[\text{"Joules"}]$$ A Joule is the metric unit energy of energy.

How much energy does it take to lift you, a ____ kg human, from the ground, to the third floor of the Ad building? (Estimate the height of the 3rd floor...)

You are probably more familiar with food "Calories" as units of energy. (Adult humans need roughly 2000 Calories / day to survive.)

  • Chemistry class: A [lower case] calorie is the amount of heat energy need to raise the temperature of 1 milliliter of water by 1${}^o$ C.
  • But on food labels: One [UPPER CASE] Calorie is actually 1000 calories of the chemistry class variety. So we alternately call one food Calorie = 1 kilocalorie (since kilo- means 1000).

Ask Google   How many kilocalories in [your Ad building calculation] joules?

Here is the map for a climb in the Tour de France up the Bisanne mountain:

If a 75 kg hiker ascends 1000 m (about 6/10 of a mile above their starting point) how much energy does he need to overcome gravity?

If a 75 kg bike racer (mass of person + bike) ascends 1000 m how much energy does she need?

What's different between the two groups?

Energy comparison

The tables shows the energy content per gram of different things. (Sometimes it's KineticE, sometimes ChemE, ....) What do you find surprising in this table?




Energy content (calories per gram)

Bullet (moving at speed of sound)0.01Butter 7
Battery (auto)0.03Gasoline10
Battery (alkaline)0.15Natural gas (methane) 13
TNT (trinitrotoluene)0.65Hydrogen 26
Modern explosive (PETN)1Asteroid @ 30 km / s 100
Chocolate chip cookies5Uranium 235 20,000,000
Coal or Ethanol (alcohol) 6Water behind a dam??

From Richard Muller, Physics for Presidents.

The quickness of energy transformation

The notion that captures this idea of how quickly work is done is...

Power = $\frac{W}{t}$ = work done / time to do it=$\frac{E}{t}$ = energy/time

SI Units 1 Joule / sec $\equiv$ 1 "watt"

Remember our comparison of the energy/gram of TNT (0.65 Cal / g) vs chocolate chip (5.0 Cal / g)? But, the energy in TNT can be released much, much faster (energy / time) than the energy in a chocolate chip cookie.

Student power

A proposal on one campus was made that the exercise machinery in their equivalent of the RFC could be hooked up to electric generators, and that power could be used to offset some of the electricity consumed on campus.

[writing assignment: download and do it with a friend].

In the student power exercise, you will climb a height $h$ equal to
$h =$ (number of steps) $\times$ (height of one step [cm]) $\times \frac{\rm 1\ m}{\rm 100\ cm}$.


Your mass is... $m$ [kg] = (weight [lbs]) $\times \frac{\rm 1\ kg}{\rm 2.2\ lbs}$.

Estimate in class before they do it...


The total gravitational energy you gained was...
GravE = $m\cdot g\cdot h=$
    $=$(your mass [kg]) $\times$ 9.8 m/s${}^2 \times$ (height [m]) = _____ Joules

So your power = energy / time is:
power = (GravE [J]) / (time [sec]) = _____ Watts

For comparison

Students are exercising on machines at the Rec Fit every day. What if we hook them up to generators...Could we power the campus?

You measured your power for a brief outburst of climbing steps.
  • Going easy for about 20-30 seconds, you averaged 176 Watts and rated it 25% of maximum effort.
  • Going hard for 15 seconds or less, you averaged 325 Watts and rated your average effort 68% of maximum.

You measured your power for a brief outburst of climbing steps:

    High effort: (Watts) 457, 186, 492 (80%), 2403, 324, 521, 380
    average(*) = 370 W

(*) It turned out that the group that got 2403 Watts of power had converted 730 cm to 72 meters. It should have been 72 meters. When that was corrected, 2403$\rightarrow$240.3 Watts, and that was used in calculating the class average. See below for the easy way to do unit conversions:

Unit conversions: The easy way...

Google: 730 cm in meters.

700 W is a typical toaster power. Here's how long an elite athlete can keep up 700 W...
Can olympic cyclist Robert Förstemann toast bread?

The olympic cyclist was able to sustain a power of 700 W--about twice our *hard going* class average--for only about 1.5 minutes.

I mentioned that in the Tour de France time trials, top competitors have been measured with powers of 400 W for 20 minutes.

Normal folks, exerting themselves gently (jogging or brisk walk or playing ultimate) might average 100 W.

Units for energy

power = (GravE [J]) / (time [sec]) = _____ Watts

or $$P\text{, Power (W)}=\frac{E \text{, Energy (J)}}{t \text{, time (s)}}$$ A little algebra: $$P * t = E$$

So we could express a Joule as 1 Watt$\cdot$second.

In general, Power * time = energy.

My household energy bill, October 2009:

We calculated that this electric energy cost \$56.33 / 470 kWh = \$0.12 / kWh, about 12 cents for a kWh.

A more familiar unit of energy is a food Calorie. It turns out that one food "Calorie" (capital C) is actually a "kilocalorie". 1 calorie (little c) is the energy needed to raise 1 gram of water by 1 degree Celsius.

In response to How many kilocalories in one kWh Google responds 840 kilocalories (food "C"alories).

A typical adult consumes 2000 Calories in a day.

2000 Calories * (1 kWh / 840 Calories) = 2.4 kWh.

So, if we were buying 2000 Calories of electrical energy it would cost $$\text{2.4 kWh}*\frac{\text{12 cents}}{kWh}=\text{29 cents}$$ Cheap!

  • What is the cost that Paul paid for 1 kWh of electricity?
  • How many kilocalories are there in 1 kWh? (Google is your friend...)
  • What do 2000 kilocalories of electricity cost?

  • How many kWh per day?
  • How kcal in that many kWh (per day)? (Google is fine...)
  • How much money per day?
  • How many "people" per day could this much energy feed? (2000 kcal needed for each person...)

Google How many Joules in a kWh?

Extra Credit assignment

These folks built a Human Powered power station (Bang Goes the Theory).

Watch that episode and respond to these questions:

  1. What were some of the Collins' daily activities that took the most energy?
  2. What suggestions for saving energy around the house occur to you, or were explicitly mentioned in the episode?
  3. What surprised you the most?

Light Bulb

light bulbHow many light bulbs could you keep going???

100 W=100 J/s for an incandescent, '100 W' bulb.

Only About 2% of the incoming ElectricE comes out as light, 98% is heat.

Luminous efficiency

In general, efficiency is: $$\frac{\text{energy output *in some useful form*}}{\text{energy input}}\left(\times \text{100%} \right).$$

candle0.04%
incandescent2%
compact fl.10%
LED (theory)40%

See this story about LED traffic lights (Elkhart County, winter of 2013).

2010: A european ban on selling incandescent light bulbs has gone into effect. They want citizens to buy compact fluorescents or LED fixtures which have a much higher efficiency. But a german businessman is trying to get around the ban by selling incandescents labelled as "heat balls".

Electric usage

Power = E/t $
\to$ Energy = Power * time.

We've been measuring energy in Joules, but many other units are possible. The electric company uses kiloWatt*hrs which typically costs about $0.13

How many Joules are there in 1 kWh?

Automobiles

Mustang

745.7 W = 1 horsepower

This 1967 Mustang had engines ranging from 115-390 horsepower.

200 horsepower $\times \frac{746 W}{1 hp} \approx 1.5 \times 10^5$ W.