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Home , Acceleration, British thermal unit, Potential energy, Units of energy

Let’s pick up where we left off last …..the topic was gravitational

Now, let’s talk about a form of energy

Imagine you are standing on top of half dome in Yosemite valley, holding a rock in your hand.

The rock has no , but if you threw it off the cliff it would have quite a bit of kinetic energy by the time it hit the valley floor.

We say that the rock has potential energy. If m is the of the rock and h the height above ground, the potential energy of the rock is…

PE = mgh What is g here?

Physics 190E: Energy & Society Physics of Energy II - 1 Fall 2007 Recall also the reading assignment….

Reading assignment in textbook - chapter 3 - , energy &

Physics 190E: Energy & Society Physics of Energy II - 2 Fall 2007 g is known as the . It measures the strength of the ’s gravitational pull on falling objects.

Galileo demonstrated that all objects fall the same way.

If two objects are dropped from the same height at the same time, then they will hit the ground at the same time (as long as other like air resistance are negligible).

Falling objects accelerate downwards at a of …

g = 9.8m / s2

Acceleration is the rate of change of with time. So, the units of are the units for velocity divided by another factor of time.

Physics 190E: Energy & Society Physics of Energy II - 3 Fall 2007 More on acceleration & related physics…

2007 Ferrari F430

Weight: 3196 lb (1450 kg) Acceleration: 0-62 mph in 4.0s Top :>196 mph (>315 km/h)

2007 Toyota Prius Economy city/highway 11/16 mpg

Let’s calculate its acceleration in meters/(second)2 0-60 mph in 10s 60mpg(city), 50mpg(hway)

Physics 190E: Energy & Society Physics of Energy II - 4 Fall 2007 Basic physics result - if an object starts at rest at time t=0 and accelerates with a constant acceleration, its velocity increases linearly with time…. v = a! t

acceleration

If we want to figure out the acceleration, we can rewrite this as a = v /t

The car accelerates, reaching a velocity of v=62 mph = 28 m/s in t=4 s, which gives A little bit smaller than the !1 !2 gravitational acceleration of a = (28ms )/(4s) = 7ms g=9.8m/s2

Physics 190E: Energy & Society Physics of Energy II - 5 Fall 2007

While we’re talking about acceleration, let’s introduce another piece of basic physics … ’s 2nd law. F = m ! a

Force equals mass acceleration. If there is a net on an object, it will accelerate. Conversely, if something is accelerating, there must be a force on it.

Back to …the gravitational force (at the earth’s surface) is F = m ! g Setting these two expressions equal, we see that the mass cancels giving

a = g This is your . The force that a scale pushes up on your feet independent of the mass of the object. with to counterbalance gravity.

Physics 190E: Energy & Society Physics of Energy II - 6 Fall 2007

The fact that the are the same in these two equations has very deep significance in physics. This “” led Einstein to his theory of gravity - - in which the gravitational force is a manifestation of the of spacetime.

The mass in Newton’s 2nd law (F=ma) is known as the inertial mass, while the mass in the gravitational force law (F=mg) is known as the gravitational mass.

The equivalence principle has been demonstrated experimentally to one part in a trillion…

Physics 190E: Energy & Society Physics of Energy II - 7 Fall 2007 Finally, back to energy

We can check that potential energy indeed has the …

PE = mgh If the mass is measured in and the height in meters then the units of potential energy work out to be…

2 2 2 kg! (ms" )! m = kg! m ! s" =

Recall these units came out naturally from the formula for Kinetic energy 1/2 mv2

Physics 190E: Energy & Society Physics of Energy II - 8 Fall 2007 We can also check that falling objects satisfy .

If we drop something from a height D at time t=0, then it’s and as functions of time are given by 1 h(t) = D ! gt 2 v(t) = !gt 2 Now, let’s calculate the total energy as a of time. 1 E = KE + PE = mv(t)2 + mgh(t) 2 The result is actually independent of time and equal to the initial potential energy, demonstrating conservation of energy. 1 1 E = (!gt)2 + mg(D ! gt 2 ) = mgD 2 2

Physics 190E: Energy & Society Physics of Energy II - 9 Fall 2007 A practical application of gravitational potential energy …… How to store energy without a battery?

We’ll see that one problem with is that it’s difficult to store. Batteries are only practical for relatively small amounts of energy. How do you store more massive quantities? One way is to use it to lift up water and convert the to gravitational potential energy. This is called pumped storage .

The Northfield Mountain pumped storage hydroelectric plant - operated by First Light Power Resources - is located in Northfield, MA about 20 minutes north of campus (up route 63).

Physics 190E: Energy & Society Physics of Energy II - 10 Fall 2007 The 1080 MegaWatt plant at Northfield Mountain facility opened in 1972 and was the largest in the world at that time.

During periods of low demand, water is pumped 5.5 from the Connecticut river to a 300 reservoir, 800 feet above the river, which holds 5.6 billion of water.

In generating mode, water flows downhill through 4 generators at a rate of 20,000 gallons per second.

Possible paper topic

Info from http://en.wikipedia.org/wiki/Northfield_Mountain

Physics 190E: Energy & Society Physics of Energy II - 11 Fall 2007

So far, we’ve talked about two forms of energy - kinetic energy and gravitational potential energy. Now, we’ll introduce a 3rd - thermal energy. We will try to understand what it means for something to be hot and how much energy it takes to something up? We’ll see that for a , like the air in this room, thermal energy is just the sum of the kinetic of the individual gas molecules. Read Chapter 7

Understanding the mechanical equivalence of heat - that could be transformed into heat and vice-versa - was a major achievement of 19th century physics.

This effort was closely tied to the industrial revolution and the need to understand how things like steam engines (which convert heat into mechanical energy) work …

Physics 190E: Energy & Society Physics of Energy II - 12 Fall 2007 A good way to start into this subject is to talk about the amount of energy it takes to heat something up?

One way to talk about this is just to give it a name ….. The (or BTU) is defined as

1 BTU = amount of energy required to raise the of 1 of water by 1 degree farenheit.

We already have another unit of energy - the Frigidaire 6000 BTU Air Conditioner . We need to know how many Joules does a BTU correspond to? This is a question Really this means BTU/ - a for experimentalists? measure of the cooling capacity of the air conditioner

Physics 190E: Energy & Society Physics of Energy II - 13 Fall 2007 This was a question that interested James Joule, himself….

Of course, at the time the mechanical unit of energy in use was not the Joule.

It was the -pound. Before coming back to Joule’s work, let’s take yet another detour into units and talk about the foot- pound as a measure of energy…. This will allow us to bring up another important point about energy. James Joule, 1818-1889

Physics 190E: Energy & Society Physics of Energy II - 14 Fall 2007 The usual definition of energy given in introductory physics textbooks is …. energy = capacity to do work

Of course, to complete the definition we need to ask what physicists mean by work?

If you sit in the library reading a book for a course, are you doing work in the physics ? No

In physics work means very specifically exerting a force through a , with the direction of in the same direction as the force.

Physics 190E: Energy & Society Physics of Energy II - 15 Fall 2007 This part about directions is important ….

An elevator does work when it takes us between floors, because the force it exerts is in the same direction as its motion - up.

However, if someone is walking along carrying something, they are not doing any work (at least on the object they are carrying) in the physics sense - there is no force in the direction of motion. The force is upwards, while the direction of motion is forwards.

This makes sense because in the first case, the elevator goes up and the work it does increases the potential energy of itself and whoever is inside. However, in carrying water, the water is always staying at the same height.

Physics 190E: Energy & Society Physics of Energy II - 16 Fall 2007 So long as the force and motion are in the same direction, the formula for work is W = (Force)(Distance) = F D

The foot-pound combines a unit of force - a pound -with a unit of distance - a foot - and is thereby a unit of work or energy. 1 foot-pound is the amount of work that must be done to raise a 1 pound weight by 1 foot. This also gives the change in potential energy of the 1 pound weight.

Note: Pounds are used both as a measure of force and of mass, which can be confusing. A pound-mass is the amount of mass that weighs 1 pound on the surface of the Earth. However, on the surface of the moon it would weigh something less than a pound…. When we make a conversion 1 pound = 2.2 kilograms, we are really talking about the pound-mass.

Physics 190E: Energy & Society Physics of Energy II - 17 Fall 2007 Yet another unit…..the SI unit for force is called the Newton.

1Newton = 1N = 1(ki logram)(meter) / (second)2

This makes sense, based on the equation F = m ! a

The unit of force is the unit of mass times the unit of acceleration.

Let’s check that work has the same dimensions as energy. Work equals force times distance. So a unit of work is a Newton-meter.

1 Newton-meter = 1 (kg m s-2) m = 1 kg m2/s2=1 Joule

Physics 190E: Energy & Society Physics of Energy II - 18 Fall 2007 Back to Joule and the mechanical equivalent of heat

Joule built an apparatus in which water was heated by mechanical agitation. He could measure both the temperature change in the water and the amount of work done by the agitator.

Recall that 1 BTU is the amount of energy needed to raise the temperature of a pound of water by 1 degree farenheit. Joule found that 1 BTU = 773 foot-pounds.

The modern measurement is 1 BTU = 778.3 foot pounds. So Joule’s measurement was quite good.

Physics 190E: Energy & Society Physics of Energy II - 19 Fall 2007 Let’s focus on this relation between heat and mechanical work 1 BTU = 778.3 foot-pounds

and note that 1 BTU is approximately the amount of energy released by burning a match. Burning releases the stored in the wood.

We see that this same amount of energy with lift a 1 pound weight nearly 800 feet in the air, or equivalently a 100 pound weight up to a height of 8 feet.

It is quite remarkable that the chemical energy stored in such a small piece of material could accomplish such a feat!

Indeed, the fact that burning fossil yields quite useful amounts of mechanical energy is what made the industrial revolution possible….. We’ll return to this in much more detail later.

Physics 190E: Energy & Society Physics of Energy II - 20 Fall 2007 Not only are the capacities of refrigerators usually stated in BTU’s. Total annual world energy usage is often stated in terms of “Quads”.

1 Quad = 1 quadrillion BTU = 1 x 1015 BTU

In 2004, total world was 447 Quads…

Physics 190E: Energy & Society Physics of Energy II - 21 Fall 2007 Back to thermal energy

Thanks to Joule we can measure the amount of energy that it takes to heat something up. Can we also understand the nature of the energy contained in a hot object via the basic laws of physics? What is thermal energy?

Matter comes in 3 basic phases - solid, & gas. The easiest to understand are and that’s where we’ll start.

In the case of simple gases, there is a simple formula relating the thermal energy of the gas to its temperature.

Solids and are more complicated

Physics 190E: Energy & Society Physics of Energy II - 22 Fall 2007 We’ll consider a gas that’s made up of single in a container. The atoms travel around in straight lines colliding occasionally with each other and with the walls of the container.

The number of atoms in a gas is immense (approximately 3x1022 in a liter container at room temperature and ).

The pressure of a gas comes from collisions of the atoms with the walls of the container. The faster the atoms in the gas are moving, the higher the pressure.

The speed of the atoms is in turn related to temperature.

Physics 190E: Energy & Society Physics of Energy II - 23 Fall 2007 The average kinetic energy of gas atoms is … m = mass of atoms 1 3 T = temperature mv 2 = k T 2 2 B where kB is known as Boltzmann’s constant and is given by

"23 kB =1.4 !10 J /K

and temperature is measured using degrees Kelvin. (This is the K in the units of Boltzmann’s constant)

The total energy of the gas is just the sum for all the gas atoms.

3 E = Nk T N = number of gas atoms gas 2 B

Physics 190E: Energy & Society Physics of Energy II - 24 Fall 2007 8 quantitative problems 1 short answer problem

To be handed in in class

Be sure to show your work on the quantitative problems. Don’t just write down the answer.

Physics 190E: Energy & Society Physics of Energy II - 25 Fall 2007