# (15) Energy

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 Kinetic

An 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 surface--yet 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 experts-only 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 t2/2

Multiply both sides by g:

gh = g2t2/2

Then since the final velocity is

v = gt

one gets

gh = v2/2

By the last equation, as the object loses elevation--assuming nothing interferes with its motion--v2 grows, and as noted, this growth does not depend on the path taken.

This exchange between h and v2 also works in the opposite direction: an object rolling up an incline loses v2 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 v2 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 + v2/2

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 + mv2/2

A well-known fact--already hinted at--is 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, mv2/ 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 mv2/2 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 Energy

Textbooks 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.

For the record, work is measured in joules, after James Prescott Joule (1818-89), a brewer in Manchester, England, whose experiments helped establish the fact that heat was a form of energy (see further below) and not some mysterious fluid permeating matter. Since any such work can be performed by a machine, one can also loosely define energy as anything that can cause a machine to turn.

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 *****

### Tidbits

Packaged foods in the US carry labels listing their nutrients and the amount of calories (actually, kilocalories) per portion and per given weight. In other countries the information may differ, and the British "Cadbury's Wispa" milk chocolate bar offers a choice of kilocalories and kilojoules:

 . 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

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