Let s pick up where we left off last time..the topic was gravitational potential energy Now, let s talk about a second form of energy Potential energy Imagine you are standing on top of half dome in Yosemite valley, holding a rock in your hand. The rock has no kinetic energy, 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 mass of the rock and h the height above ground, the potential energy of the rock is PE = mgh What is g here? Physics of Energy II - 1
Recall also the reading assignment. Reading assignment in textbook - chapter 3 - work, energy & power Physics of Energy II - 2
g is known as the gravitational constant. It measures the strength of the Earth 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 forces like air resistance are negligible). Falling objects accelerate downwards at a rate of g = 9.8m / s 2 Acceleration is the rate of change of velocity with time. So, the units of acceleration are the units for velocity divided by another factor of time. Physics of Energy II - 3
More on acceleration & related physics 2007 Ferrari F430 Weight: 3196 lb (1450 kg) Acceleration: 0-62 mph in 4.0s Top Speed:>196 mph (>315 km/h) Fuel Economy city/highway 11/16 mpg 2007 Toyota Prius Let s calculate its acceleration in meters/(second) 2 0-60 mph in 10s 60mpg(city), 50mpg(hway) Physics of Energy II - 4
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 = (28ms!1 ) /(4s) = 7ms!2 A little bit smaller than the gravitational acceleration of g=9.8m/s 2 Physics of Energy II - 5
While we re talking about acceleration, let s introduce another piece of basic physics Newton s 2nd law. F = m! a Force equals mass times acceleration. If there is a net force on an object, it will accelerate. Conversely, if something is accelerating, there must be a force on it. Back to gravity 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 independent of the mass of the object. This is your weight. The force that a scale pushes up on your feet with to counterbalance gravity. Physics of Energy II - 6
The fact that the masses are the same in these two equations has very deep significance in physics. This equivalence principle led Einstein to his theory of gravity - general relativity - in which the gravitational force is a manifestation of the curvature 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 of Energy II - 7
Finally, back to gravitational potential energy We can check that potential energy indeed has the units of energy PE = mgh If the mass is measured in kilograms and the height in meters then the units of potential energy work out to be kg! (ms "2 )! m = kg! m 2! s "2 = Joules Recall these units came out naturally from the formula for Kinetic energy 1/2 mv 2 Physics of Energy II - 8
We can also check that falling objects satisfy conservation of energy. If we drop something from a height D at time t=0, then it s position and velocities as functions of time are given by h(t) = D! 1 2 gt 2 v(t) =!gt Now, let s calculate the total energy as a function of time. E = KE + PE = 1 2 mv(t)2 + mgh(t) The result is actually independent of time and equal to the initial potential energy, demonstrating conservation of energy. E = 1 2 (!gt)2 + mg(d! 1 2 gt 2 ) = mgd Physics of Energy II - 9
A practical application of gravitational potential energy How to store energy without a battery? We ll see that one problem with electricity 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 electrical energy to gravitational potential energy. This is called pumped storage hydroelectricity. 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 of Energy II - 10
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 miles from the Connecticut river to a 300 acre reservoir, 800 feet above the river, which holds 5.6 billion gallons of water. In generating mode, water flows downhill through 4 turbine generators at a rate of 20,000 gallons per second. Possible paper topic Info from http://en.wikipedia.org/wiki/northfield_mountain Physics of Energy II - 11
Thermal Energy 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 heat something up? We ll see that for a gas, like the air in this room, thermal energy is just the sum of the kinetic energies of the individual gas molecules. Read Chapter 7 Understanding the mechanical equivalence of heat - that mechanical energy 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 of Energy II - 12
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 British Thermal Unit (or BTU) is defined as 1 BTU = amount of energy required to raise the temperature of 1 pound of water by 1 degree farenheit. We already have another unit of energy - the Joule. We need to know how many Joules does a BTU correspond to? This is a question for experimentalists? Frigidaire 6000 BTU Air Conditioner Really this means BTU/hour - a measure of the cooling capacity of the air conditioner Physics of Energy II - 13
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 foot-pound. Before coming back to Joule s work, let s take yet another detour into units and talk about the footpound as a measure of energy. This will allow us to bring up another important point about energy. James Joule, 1818-1889 Physics of Energy II - 14
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 sense? No In physics work means very specifically exerting a force through a distance, with the direction of motion in the same direction as the force. Physics of Energy II - 15
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 of Energy II - 16
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 of Energy II - 17
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 m 2 /s 2 =1 Joule Physics of Energy II - 18
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 of Energy II - 19
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 chemical energy 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 fuels 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 of Energy II - 20
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 10 15 BTU In 2004, total world energy consumption was 447 Quads Physics of Energy II - 21
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, liquid & gas. The easiest to understand are gases 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 liquids are more complicated Physics of Energy II - 22
We ll consider a gas that s made up of single atoms 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 3x10 22 in a liter container at room temperature and pressure). 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 of Energy II - 23
The average kinetic energy of gas atoms is 1 2 mv 2 = 3 2 k BT m = mass of atoms T = temperature where k B is known as Boltzmann s constant and is given by k B =1.4!10 "23 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. E gas = 3 2 Nk B T N = number of gas atoms Physics of Energy II - 24
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 of Energy II - 25