Forces in 2D Legacy Problem #26 Guided Solution

This is a very difficult problem, and one that is all about visualizing and reasoning. What we have is a flume ride, a water ride at an amusement park, Walt Disney World's Magic Kingdom. It's a ride in which riders get in a boat, and they plunge down a steep incline that has filled with water into a little watered area at the bottom. And what we wish to do is determine the acceleration of the boat and the riders, and what makes this a difficult problem is there's no mass given for you. So you need to do a good deal of reasoning and strategy plotting in order to get from the given information to the unknown information. And the given information is that the coefficient of friction between this boat and the water surface that it rides along is 0.12, and the angle of incline is 43 degrees. There's a suggestion given, a strong suggestion, begin with a free body diagram, draw the forces acting upon the boat. So what we have is a gravity force pulling straight down on the boat. It has components parallel and perpendicular to the inclined plane, which becomes a matter of greater interest to us. And then there's the friction force, which goes up the inclined plane and parallel to it. And then there's the normal force, which is perpendicular to the plane and pushing up on the boat. Now, since we don't have the mass, what we're going to have to do is approach the problem using simply variables. For instance, for the F perpendicular, I can't calculate a number for it since the mass is not known. But what I can say about it is that the perpendicular component of the weight vector is mg cosine theta. So I'm going to write it down as such, mg cosine theta. And the normal force balances it, so it is mg cosine theta. Now, the friction force is based upon the value of the normal force. Since we don't have a value for the normal force, we're going to have to simply express the friction force in terms of variables. So for the friction force, it's simply mu F norm, where the F norm was already stated as mg cosine theta. So for F friction, I should write it's equal to mu times m times g times the cosine of theta. Finally, we have the parallel component of the gravity force, and that is mg sine theta. It's the parallel component of gravity that brings the boat along the incline, accelerating it down the incline plane. And it's friction which opposes it. So when it comes time to write the net force equation, I would write the F parallel minus the F perpendicular. We're going to do that now. We're going to write F net equal ma, and for F net, we're going to write F parallel minus F perpendicular. And together, that's equal to ma. Now for the F parallel minus F perpendicular, I'm going to substitute my expressions in for those two variables into that equation, so that the equation now becomes mg sine of theta minus mu mg cosine theta is equal to ma. That was just simply writing F parallel, mg sine theta, minus F friction, which is mu mg cosine theta, and setting it all equal to ma. Now you'll notice in our equation, we have an m in every term, so we can divide through by m and cancel that, and the equation becomes g sine theta minus mu g cosine theta is equal to a. So to calculate the acceleration, you can see you need to know three things. g, which is 9.8, theta, which is given here as 43 degrees, and mu, which is given here as 0.12. We now have sufficient enough information to solve this problem. Go ahead and plug that information into the equation and solve for acceleration as g sine theta minus mu g cosine theta. When you're done, you should get 5.8 meters per second per second, at least rounded to two digits, and that would be the answer.