Forces in 2D Legacy Problem #25 Guided Solution

This is a very difficult problem, one which is all about visualizing, thinking, and strategy planning. You'll use your calculator very little on this problem. In fact, there's hardly any numbers that are provided in order to use your calculator. I love these types of problems. But we're going to begin with a little bit of an experiment of your own. If you could get a book with a cover on it, maybe your physics book, place it up on your desktop and put your calculator on top of your book. Then slowly and gradually take the cover of that book and begin to elevate it. You could tilt the whole book, but it's kind of nice that the cover is hinged at one side. You can just simply begin to elevate one of the covers. What you're doing is you're making an inclined plane out of the cover of your book with your calculator on there. Now, you'll notice at a very small angle, the calculator remains on the book. But as you begin to raise the cover up more and more and elevate it more and more, the inclined angle increases. Keep doing that and eventually, kerplunk, there goes the calculator sliding down the cover of your book. Now, at the point just before the calculator budged from rest, what you had is a situation in which the calculator's friction force had reached a maximum and was balancing the parallel component of gravity. The condition at which this happens is the condition that Anna and Noah are investigating. They're trying to find the maximum angle of incline that keeps the block up on their inclined plane. And they find it to be 38 degrees. You could say that at 37.999 degrees, the F parallel is equal to the F friction. Now, it's the force of static friction, which can range anywhere from zero to some maximum value, the maximum value of which is equal to mu times F norm. So the way I'm going to approach this problem is I'm going to draw a diagram, a free-body diagram, and I'm going to draw the forces that are acting up on our calculator or our block or whatever it is on the incline. There's gravity straight down. There's normal force perpendicular or normal to the incline. And there's friction force that opposes the would-be motion of your calculator or your block down that surface. Now, what we know is that the F parallel is equal to the F friction. We're going to try to figure out what mu value would give you F parallel equal to F friction when you have a 38 degree angle. So what I'm going to do is I'm going to start out by saying F parallel equal F friction. I'm going to follow along and even write some of this stuff down. Now, for F parallel, I can write mg times the sine of theta after all. That's how I'd calculate it. Now, you're going to find it's impossible to actually calculate this because the m value is not given. Now, I'm going to write for F friction, mu times F norm. Then I'm going to write it again as mu times mg cosine theta. Now, I do that because the F norm value balances the perpendicular component of the weight vector and the perpendicular component of the weight vector is just mg cosine theta. So now I've changed my F parallel equal F friction equation into this form. mg sine theta equal mu mg cosine theta. Wow, now you notice there's an m and a g on both sides and I love it when this happens. You can cancel the m's out by dividing through by m and you can cancel the g's out by dividing through by g's. And the equation now becomes sine theta equal mu times cosine theta. Hey, this is one equation and one unknown. We know what mu is and we ought to be able to solve therefore, or we know what theta is so we ought to be able to solve for mu. If you want you can do one more algebra step and divide each side by cosine of theta in which case you have sine theta over cosine theta also known as the tangent of theta and that's equal to mu. You didn't have to do it that way but I always find it kind of elegant to get the equation in the form of the tangent of the angle is equal to mu. Now if you're trying to find mu just plug 38 degrees in and take the tangent of that value and you get 0.7813 and that's coefficient of friction for this situation in which 38 degrees starts the object budging from rest and sliding down the surface.