Sometimes you think you have a complete understanding of something and then BOOM – a simple problem throws everything out the window. Consider a very simple physical problem in which a block is pushed with a frictional force. Such issues are common in your introductory physics textbook, but they often lack some subtle details.
I address two basic ideas in physics: the principle of momentum and the principle of working energy. Let's use these two ideas for some simple physical cases and see what happens. It will be fun.
The principle of momentum implies that a net force on an object is equal to the change in momentum (p) divided by (t), the change of time (the time rate of momentum change). Oh, momentum (for most objects) can be defined as the product of mass (m) and velocity (v). I'll show you this using a one-dimensional example so I can avoid using vector notation (this will keep it simple). Here's the momentum principle (in 1
Now let's use this. Suppose I have a car with very low friction and constant compressive force (in this case a fan is mounted at the top). Since there is a force, the car gets faster. That's how it looks.
We can now use the Pulse Principle to determine the speed change over a period of time. Here are some mostly real values for the above mentioned car (I have made some minor changes due to measurement errors) Force and time interval I get a momentum change (F dt) of 0.45 kg m / s. If I divide this momentum change by mass, I get a final velocity of 0.53 m / s (assuming it starts at rest). Yay.
Okay, let's do it again. This time with TWO fans. Here is a cart with two equal forces pushing in opposite directions. After switching on the two fans, I push the car to the right.
In this case, the net force on the car is zero Newton, since the right pushing force has the same strength as the left force. At a net force of zero, the momentum does not change and the carriage moves at a constant speed.
Another case. Suppose I take a box of some mass and pull it along the table at a constant speed. In this case, a force pulls to the right (the string) and a frictional force to the left.
Since the net force is zero, the momentum does not change. Everything is OK.
That's not entirely new. In fact, you can derive this idea from the principle of momentum. The work-energy principle states that the work (w) at a point mass is equal to the change in kinetic energy. The work is done by a force that moves a certain distance. Actually, it depends only on the force in the direction of movement. As an equation, it looks like this.
Here θ is the angle between the force and the displacement. When the force pushes backwards, you can have negative work. The kinetic energy depends on the mass and the speed.
Okay, let's go back to the fan car from the top. Suppose I want to look at this problem using the working energy principle instead of the pulse principle. In this case, I need an extra thing – the distance over which the force is applied. The same fan video shows that the force pushes the car over a distance of about 0.79 meters. Now I can calculate the work (the angle is zero degrees) with a value of 0.11 Joule. If I set this equal to the final kinetic energy, I can dissolve to the final velocity and get 0.528 m / s. Boom. This is essentially the same as the impulse principle.
What about the two fans that push in opposite directions? In this case a fan does some work – let's just say it makes 0,11 Joule. The other fan has the same force for the same distance, but it pushes in the opposite direction. For the backward pushing force, the angle between the force and the displacement is 180 degrees. Since the cosine of 180 degrees is negative 1, the work done by this force is -0.11 Joule. This means that the total work equals zero joules and the kinetic energy changes by zero Joule. This can only happen if the car moves at a constant speed. Great.
What about the block being pulled over the table with friction? Again, the two forces are the force of the string pulling to the right and the friction pulling to the left. The total work on the block would be zero and it would move at a constant speed.
BUT WAIT! There is a problem. What if you measure the temperature of this block before and after drawing? Here are two thermal images – I also put a piece of Styrofoam on the floor so you can see the temperature change.
It's not a big temperature rise, but it's actually warmed up. If I move the block a greater distance (or back and forth), you can see a light streak on the surface. This is an area where the temperature of the table is rising – the block also gets hotter.
But as the block warms up, it means the energy is increasing. In this case, it would be an increase in heat energy. How can the block increase in energy if no work is done on the object? That is a mystery indeed. How is it possible that there is zero work AND an increase in energy.
Here is the answer. You can see that in another example. Suppose I rub two brushes together instead of a block and a table. Watch what happens.
Note that when pulling the brush, two forces act. My hand works (positive work) and the brushes work (negative work). But watch carefully. Note that the brushes bend when the brush (and my hand) moves to the left a certain distance. This means that the force exerted by the lower brush on the upper brush moves a shorter distance than the hand. Even if the brush force is the strength of my hand, the brush will do less as it moves a shorter distance. That is, the total work on the brush is NOT zero Joule, but a positive amount.
Of course, the brush is an analogy for friction. We like to see friction as this nice and simple interaction, but that's not it. For the block sliding on a table, the frictional force is an interaction between the surface atoms in the block and the surface atoms on the table. It is not so easy. Physics textbooks like to treat a block as a point object – but it is not a point object. It is a complicated object of countless atoms. In the case of friction, you can not forget that and just treat a block as a point object. It does not work.
Let's just be clear. If you are asked in a physics textbook to calculate "work by friction," just say "no." Just say no. You can not really calculate that. Yes, we want to make physics as simple as possible – but not so easy that you get into impossible situations, like with a block that slides at a constant speed.
Oh, but wait. There are a number of physics textbooks that ask for friction after work. The first book I snatched had an example that looked something like this:
Jake pulls a box weighing 22 kg. The rope forms an angle of 25 degrees to the horizontal. The kinetic friction coefficient is 0.1. Find the work of Jake and the work of friction in case the box moves 144 meters above the ground.
bad. Bad question. You could calculate the frictional force – but you can not calculate the work done (unless you know something about the changes in heat energy). If you calculated the friction work as a friction force multiplied by the moving distance of the block, how would you explain the increase in thermal energy of the block (and the bottom)? Oh, but you could solve this problem with the momentum principle and it would not be a problem. Remember that the momentum principle deals with forces and TIME, not DISTANCE. Although the frictional force acts over a different distance, the time for the frictional force and the force pulling on the string is the same.
What should we do then? If we can not work smoothly, how should we teach physics? Here is the problem. The main goal in physics is to build models that match actual experiences. These models could be a big idea, like the principle of working energy – and that's great. Consider an example with another model. What about a globe? It is a model of the earth. It even shows the position of the continents and everything. But what if I want to use this globe and measure its mass and volume so I can determine the density of real Earth (in full size)? That would not work, because the globe is actually not Earth. The same applies to the working energy principle. It's great for some things, but you can not just use it wherever you want.
Finally, let me point out that I know about these problems with work and friction only because of my good colleagues Bruce Sherwood and Ruth Chabay (yes), the authors of my favorite textbook on physics Matter and Interactions ). This happened during an informal side talk at the recent meeting of the American Association of Physics Teachers (AAPT). Honestly, there are so many educators at this conference who have a tremendous impact on the way I think about physics. It's always great to see her.
Other great WIRED stories