Work and Power
Up to now, we have discussed forces and motion. Now, we should probably work on better-quantifying things that involving forces and motion. So, we introduce the concept of work.
We'll worry about more accurately pinning down energy later; for now, we can just worry about work.
Start with this scenario: take an object, exert a force on it, and move it through a distance:
work: the amount of energy transferred by exerting a force through a distance.
The way we verbally say this would be, "[thing exerting the force] has done work on the object." The mathematical expression, then, should take this general form:
Where is used to describe the path along which an object travels, which we often simplify to something like .
The " " symbol is the symbol for the dot product. For a better understanding of the concept of what a dot product is, refer to the section on dot products and cross products. For the purposes of our course, though, we will simply take the definition of a dot product and arrive at the formula we will actually apply:
Notice that the was replaced with a , and the vector symbols have been removed. With the vector hats gone, this means we simply use the magnitudes of those vectors in our formula, and the distance the object has traveled.
An example diagram involving work resulting in this more general equation may resemble:
Now, what sort of units does work have? As noted above, since it is a force times a
displacement, we have . We can abbreviate this as , or Joules, which is a unit for energy. We will discuss energy in more detail, but for now, we can make a mental note that work must somehow be somehow related to energy.
Now, what happens if we do work vertically?
Here, the force is exerted directly in the same direction as the vertical motion. In order to lift the object, the force must be equal to the force of gravity. Thus, the work done to lift the object is:
The angle here is and not because, remember, this is the angle between the two vectors, not the angle either makes with respect to the horizontal.
If we call the distance it moves off the ground in place of , we get:
Now, a question regarding this theory: we have satellites in orbit around Earth, like so:
Assume that it has a constant orbital speed. Does the force of gravity do work on this object? Why, or why not? (hint: recall which way centripetal forces point!)
Next, we can begin to talk about power:
power: the rate at which work is done.
So, more mathematically, if we were find the rate at which work is done, we would get this equation:
This has units of , which you may know as , or Watts. The rating on light bulbs is also in Watts, as this is the amount of energy used by the bulb in the form of light every second. (e.g. a 100-W light bulb uses 100 J of energy every second.) So, the expanded version of Power would be:
So, as an example, if you found that you did 100 J of work on a box and it took you 5 s to displace the box, you have developed 20 W of power.
Attempt the following:
1) An elevator motor can lift its car and any passengers in it at a rate of ~5 s per floor. Suppose that the mass of the car and its passengers is ~2000 kg. Find the amount of power developed by the motor in moving from the ground floor to the fourth floor.
2) A warehouse worker pushes a 50 kg box 10 m across the floor with a force of 100 N, then lifts the box up 2 m onto a platform. If all of this transpired in 10 s, what amount of power did the worker develop?
3) A winch pulled a Jeep off a patch of ice that was 50 m long. If it took 120 s to do this, and the motor in the winch can develop 1000 W of power:
a) find the force that must have been exerted on the Jeep.
b) How much more power would the winch have to develop if the impatient Jeep owner wanted his Jeep back in under one minute?
Energy
Up to this point, we have talked about forces, motion (be it motion in straight lines or in circles) and work. We can, however, go one level deeper: energy.
Energy is probably not a very intuitive concept for most; at its core, it is a quantity that arises by the nature of interactions of atoms and molecules. For the next few units, we will mostly focus on energy as a quantity one object has, and treat that as the sum of the energies of every atom and molecule making up that object.
We're going to start by defining a new term, potential energy:
potential energy: the energy an object has as a result of its position, shape, or composition. (units: Joules)
The first two conditions in the definition- position and shape- pertain to different aspects of physics. Potential energy in the first case is measured with respect to a reference position. We can think of lifting an object and thus changing its potential energy. For the second case- shape- we can think of compressing a spring, or drawing back a bowstring, which gives the
spring/bowstring potential energy.
As we discussed in the section on work, there is a certain amount of work done to lift an object vertically against gravity:
We can realize that by doing work on an object, its potential energy has changed. So, more comprehensively:
The second term we need to define is kinetic energy:
kinetic energy: the energy an object has as a result of its motion. (units: Joules)
The mathematical formula for this needs to be derived. Consider the formula:
(*)
This formula describes an object's change in velocity through a distance. What is causing this acceleration? A force, of course! We know from Newton's Second Law that:
Solve for :
You may recall that . The left-hand side of the equation suggests a change in
something, because we have a final term minus an initial term. If we define the kinetic energy of an object to be , we arrive at:
This is the work-kinetic energy theorem, which states that the work done on a moving object changes its kinetic energy.
Based on this and the previous idea about work and potential energy, we can realize something deeper about work: it is the transfer of energy between two objects!
Let us expand this realization into something more formal. Consider an object released from a height and falling through a distance, and studying how its energy changes. Its potential energy goes as:
Where is the work done by gravity to move the object through that vertical distance. Because the object is falling, we can consider this negative:
We also know that its velocity will change as a result of this work done, so its kinetic energy changes:
Because the two changes in energy are equal to , we set them equal to each other:
Move the initials to one side, the finals to the other side, and we get:
Or, put another way:
This is the Law of the Conservation of Mechanical Energy. In words, it states that energy can neither be created, nor destroyed; only transformed from one form to another. The only way to get more energy into a system would be to do work on the system. So, the general law looks like:
