
Index 12b. Orbital Motion 12c. Venus transit (1) Newtonian Mechanics 13. Free Fall 14. Vectors 15. Energy 16. Newton's Laws 17. Mass 17a. Measuring Mass in Orbit 17b. Inertial balance 18. Newton's 2nd Law 18a. The Third Law 18b. Momentum 18c. Work 
Potential and KineticAn interesting thing about the final speed of an object descending (with no friction) from some given height h, along a sloping surface: one can change the slope, one can even change the shape of the surfaceyet the final speed v with which it reaches the bottom will always be the same. If it were not for friction, any skier gliding down a snowy hill from the summit to the base should arrive with the same speed (though not necessarily in the same time!), whether the path taken is the easy beginner's slope or the steep expertsonly route. Reducing the slope of the surface reduces the acceleration a, but it also lengthens the time of descent, and these two variations cancel, leaving the final speed unchanged. The same speed is also obtained if the object falls vertically from that height h, and in that case it is easily derived, as follows. The duration t of the fall is given by h = g t^{2}/2 Multiply both sides by g: gh = g^{2}t^{2}/2 Then since the final velocity is v = gt one gets gh = v^{2}/2 
By the last equation, as the object loses elevationassuming nothing interferes with its motionv^{2} grows, and as noted, this growth does not depend on the path taken. This exchange between h and v^{2} also works in the opposite direction: an object rolling up an incline loses v^{2} in direct proportion to the elevation h it gains. A marble rolling down the inside of a smooth bowl gathers speed as it approaches the bottom, then as it shoots up the other side it loses all of it again. If no friction existed, it would rise again to the same height as the one from which it had started. A simple pendulum, or a child on a swing, also trades height for v^{2} and back again, in the same manner. And bicycle riders are well aware that the speed gained rolling down a hillside can be traded for height when climbing the next slope. It is as if height gave us something with which we could purchase speed, and which later, if the occasion demanded, could be converted back to height. That "something" is called energy. It was already briefly discussed in an earlier section. This back and forth trading suggests that perhaps the sum gh + v^{2}/2 has a constant value: if one part decreases, the other part must get bigger. Is that sum the energy? Not quite. The effort of getting heavy load up a height h is greater than that of lifting a light one. Let us now call the amount of matter in an object its "mass. " It is obviously proportional to the object's weight, but as will later be seen, the concept of mass is more complicated than that. If energy is to measure the effort in lifting a load, it should also be proportional to its mass m. We thus multiply everything by m and write Energy = E = mgh + mv^{2}/2 A wellknown factalready hinted atis that in a system which does not interact with its surroundings, the total energy (denoted here by the letter E) stays the same ("is conserved"). In a pendulum at the extreme point of its swing, v = 0 and therefore the second term above vanishes, while the first term is at its biggest. Then as the mass descends descends, mv^{2}/ 2 increases and mgh drops, until at the bottom of the swing the first term is at its smallest and the second reaches maximum. On the upswing the process reverses, and the sequence is repeated for every swing. Both terms in the equation above have names: mgh is the potential energy, the energy of position, and mv^{2}/2 is the kinetic energy, the energy of motion. The exact number representing E will obviously depend on where h is measured from (the floor? sea level? the center of the Earth? ). Different choices are possible, and each leads to a different value of E: the formula is thus meaningful only if a certain reference height is chosen where h=0. Other Kinds of EnergyTextbooks define energy as "the ability to do work" and they define work as "overcoming resistance over a distance". For instance, if m is the mass of a brick, the force on it is mg and lifting it against gravity to a height h, against the pull of gravity, requires the performance of work W, with W = mgh Dragging that brick a distance x along level ground against the force of friction F similarly requires the performance of work W = Fx The above two types of work are discussed again, in more detail, in section 18c: Work. A third type is treated in the following section #18d: Work Against an Electric Force: The Van De Graaff generator. That section also covers the generation of lightning and the clinging of projector transparencies after they emerge from a copying machine. 
Devices or processes that convert energy from one form (column) to another (row)  

  Kinetic  Potential  Heat  Light  Chemical  Electric 
Kinetic  *****  Pendulum  Rocket Nozzle  Solar sail  Muscles  Electric motor 
Potential  Pendulum  *****  Steam boiler  x  x  Elevator winch 
Heat  Friction  x  *****  Solar heater  Fire  Electric stove 
Light  x  x  Lightbulb, Sun  *****  Firefly light  Light emitting diode 
Chemical  x  x  Quicklime kiln  Green plants  *****  Car battery 
Electric  Windmill power  Hydroelectric power  Thermocouple  Solar Cell  Flashlight battery  ***** 
.  Bar  100 gr. 
Energy Kj  885  2300 
..........Kcal  210  550 
Protein  2.7 gr  7.1 
Carbohyd.  20.8  53.9 
Fat  13.2  34.2 
Questions from Users: *** Can kinetic energy be converted to work? *** Energy from the Earth's Rotation? *** Teaching about energy in 8th grade 
Next Stop: #16 Newton and his Laws
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 11 May 2005
Reformatted 24 March 2006