Forces in 2D Legacy Problem #12 Guided Solution

An effective problem solver has a habit of reading the problem carefully and developing a mental picture of what's going on, writing down the known information and organizing it, and identifying the unknown information, and then before ever using a calculator, plotting out a strategy as to how to get from the known to the unknown information. The best means of doing that for a problem such as this is to use a free body diagram to identify the force values in their direction and to make an effort to do a horizontal and vertical analysis of the forces. So here, what I know is that lawnmower is pushing down upon a broom to clean the driveway and the downward and rightward force on the broom, the F applied, is 56.8 Newtons. And that's at an angle of 54.7 degrees with the horizontal. So F F equals 56.8 Newtons and the theta equals 54.7 degrees. The mass of the broom is 1.05 kilograms, that's M, and the coefficient of friction, sometimes called mu, between the driveway and the broom bristles is 0.567. That's mu in the equation, F friction equals mu F norm. I am to determine the rate of acceleration, A is the unknown quantity. To determine A, I know that I first must determine the F net. In determining the F net, I'll have to add all the individual forces. I'll have to know the values of the individual forces and add them up as vectors. If we're accelerating the broom across the driveway, then we're accelerating it to the right. The vertical forces must balance, and the net force would be a horizontal force. So to determine the net force, I'll need to know the F friction force, and the only way to know the F friction force is to know the normal force. So my strategy will involve doing a vertical analysis of all the forces so that I can find the normal force, then using it to calculate the friction force, and then subtracting that from the horizontal component of the applied force in order to determine F net. Here we go. My first thing I'm going to do is I'm going to take this 56.8 newtons, and I'm going to resolve it into two components, into the horizontal component and the vertical component. You'll notice on the diagram that the horizontal component is the side adjacent to 54.7 degree angle in our little force triangle. So I'm going to go 56.8 newtons, the applied force, times the cosine of 54.7 degrees, and I'm going to find the horizontal part of that applied force. It comes out to be 32.8223 newtons. That's more digits than you need, but I drowned at the end. So write down the whole thing, 32.8223 newtons. Now to find the y component of this applied force, I need to use the sine of 54.7 degrees. After all, the Fy value is simply the side opposite the 54.7 degrees, and I know the hypotenuse of this force triangle. So to find Fy, I go 56.8 newtons times the sine of 54.7 degrees, and then I get 46.3566 newtons. Write down all the digits. You're going to need to use this number later. Now I'm trying to find F-norm. That's part of my strategy. And to do that, I have to presume here that the vertical force is balanced. So the F-normal is equal to the two down forces, the F-grab plus the Fy. The F-grab can be calculated as mg. F-grab is 1.05 times 9.8, or 10.29 newtons. So the F-normal is equal to the F-grab plus the Fy, or the 10.29, plus the 46.3566. That comes out to be an F-normal of 56.6466 newtons. Now F-friction is what I need to find next. F-friction is simply mu, 0.567, times the F-norm I just calculated. Friction comes out to be 32.1186 newtons. Now to determine the F-net, I have to analyze the horizontal forces. I have two of them. They're very close in magnitude. One of them is the Fx value of 32.8223, and the other is the F-friction value of 32.118. When I add these two up, treating them as vectors, I get 0.7037 newtons. That's the F-net. If I take F-net and divide by the mass of 1.05, I'll get the acceleration. It comes out to be 0.6702. I can round that to three digits. That's 0.670 meters per second per second.