My intro physics text in my undergraduate education was by Halliday, Resnick, and Walker. I personally thought it was a quite good text, but what particularly stood out was its obsession with penguins. Probably a dozen or more problems involved penguins in very improbable situations. It was pretty funny. But there’s another problem I remember that involved a pig, which is a nice thing for undergrad physics students to think about. I won’t quote it directly as I don’t have the book with me (and I’m not sure how copyright works for physics problems anyway), but here’s the gist:
A pig slides down a ramp which is inclined at an angle 45 degrees above the horizontal. It takes twice as long to slide to the bottom of the ramp as it would have if there were no friction. What is the coefficient of friction?
It’s an interesting problem because it doesn’t give the length of the ramp, the time of the slide, or anything like that. You have to think a bit about how to approach it. In fact, let’s also pretend we don’t know the angle, and just call it θ. So let’s call the unknown length of the ramp x and we’ll figure out how long it takes to slide down without friction. We know that the acceleration is g sin(θ). Therefore the distance the frictionless pig slides in a time t is
Which implies that the time required to slide a distance x is
Well and good. But what about the case with friction? Well, aside from the force down the ramp from gravity, there’s a frictional force pointing upward. From the definition of the coefficient of friction μ = (friction force)/(normal force), we see that the total acceleration acting on the pig down the plane of the ramp is
You don’t see? Think of it in terms of forces, and then divide by m to find the acceleration. Notice that m doesn’t appear in any of the acceleration expressions. Thus the sliding time has nothing to do with how heavy the pig actually is. Plug that into our equation for sliding time, and we get that the sliding time for the pig with friction is
And that’s (we’re told), double the time of the frictionless time. Thus
Ugly, but all we’re after is μ. So solve for it!
Notice the x cancels, therefore the length of the board is irrelevant as well. Interestingly, this result also means that a smaller angle requires only a small frictional force to cause the sliding time to be twice as long. But this is what we expect, because a small angle means it will be sliding longer in the first place, giving friction a longer time to act. A silly problem to be sure, but one that’s an interesting and subtle exploration of some basic introductory classical mechanics!