Work
1. Work is a transfer, not a possession
Work is the amount of energy moved into or out of a system by a force exerted on that system over a distance. That is the whole definition, and the important word in it is moved. Work is not something an object has. A system can have energy; it cannot have work, any more than a bank account can have a transfer.
The everyday word gets in the way here. Holding a heavy box still is tiring and is not work in this sense, because nothing moves. Carrying that box at a steady height across a room is also not work done on the box by your hands, because the force is up and the motion is sideways. Your muscles are certainly spending energy, and none of it is reaching the box.
2. Only the part along the motion counts
For a constant force on a straight path, the work is the component of the force along the displacement, multiplied by that displacement.
W = F∥ d = F d cos θθ lies between the force and the displacement
That last line is where marks are lost. The angle is measured between the force arrow and the displacement arrow, and not between the force and the ground. On a ramp those are different angles, and using the wrong one gives a wrong answer that looks entirely reasonable.
One block, one displacement, one force of fixed size. Sweep the angle and watch the sign of the work change. Notice where it passes through zero, and notice that the force never got any smaller.
3. When the force changes along the way
The formula above assumes the force is the same all the way along. When it is not, the work is the area under the graph of the parallel force against displacement. Area above the axis counts positive, area below counts negative.
Three force profiles over the same 4.0 meters. The shaded region is the work. Read the area off the picture before you look at the number underneath, because on the exam the picture is often all you are given.
4. Conservative and nonconservative
Some forces do not care which route you took. The work gravity does on a book carried from the floor to a shelf is the same whether you lifted it straight up or walked it up a long ramp, and it comes to exactly zero if the book returns to the floor. Forces like this are called conservative, and only they get a potential energy.
Friction is not like that. It opposes the sliding at every instant, so the longer the path the more it takes, and a round trip does not come back to zero. Friction and air resistance are the two nonconservative forces this course asks about.
Two routes from the same start to the same finish. The direct one is short, the scenic one is long. Watch what gravity charges for each, and what friction charges.
Because friction charges by the length of the path, the energy it takes out is the friction force multiplied by the path length, never by the displacement. A block that slides out and comes back has a displacement of zero and has still warmed the floor the whole way.
5. The work-energy theorem
Add up the work done by every force on an object and you get the change in its kinetic energy.
ΔK = Σ Wevery force, with its own sign
This is the fastest route from forces to speeds, and it is the reason energy methods are worth learning. It says nothing at all about how long anything took, because no clock appears anywhere in it. For a time you still need Unit 2.
Check yourself
1. A 50 N force at 37° above the horizontal drags a box 4.0 m along level ground. Find the work done by that force, and the work done by the normal force.
W = 50 × 4.0 × cos 37° ≈ 160 J for the rope. The normal force is vertical and the motion is horizontal, so its work is 0, however large it is.
2. A block slides 3.0 m out and 3.0 m back across a rough floor, with a friction force of 5.0 N. Find the energy dissipated by friction and the work done by gravity.
Friction: 5.0 × 6.0 = 30 J taken out, because it charges by the 6.0 m of path. Gravity: 0, because the height never changed. That one experiment separates a conservative force from a nonconservative one.
3. A force rises steadily from 0 to 20 N over the first 3.0 m of a push. Find the work it does.
The graph is a triangle, so the work is ½ × 20 × 3.0 = 30 J. Half the peak force times the distance, which is also where the one half in the spring energy comes from.
4. A 3.0 kg block starts at 2.0 m/s and has 24 J of net work done on it. Find its final speed.
½ × 3.0 × v² = ½ × 3.0 × 2.0² + 24, so v² = 20 and v ≈ 4.5 m/s. Notice how little the algebra cares about what the forces actually were.
Topic 3.3, Potential Energy. Work by a conservative force can be banked rather than spent, and the account it goes into is the potential energy of a system.