 We're going to be looking at the forces acting on an object in equilibrium. And the object is going to be this garbage gobbler truck on this inclined plane. The plane is inclined at an angle of 30 degrees with the horizontal. There's also a bumper back here which keeps the truck from falling backwards. Now these strings and these pulleys we're going to find out later what they're for, but for now we're just going to ignore those. Let's take a close look at the forces that are acting on the truck in this situation. First we know that because it's in equilibrium that the force is acting on it at up to zero. There are three forces. Those forces are the weight of the truck downward pointing to the center of the earth. The bumper exerts a force parallel to the plane pointed like this. And the plane exerts a force perpendicular to the plane like this. That's a normal force. Those three forces at up to zero. Now let's take a closer look at those forces and how they depend upon the angle of the plane. And for that I'm going to use this setup because it's a little bit smaller and I can move it around. So here's the bumper and here's the plane and here's the car instead of the truck. In this situation where the plane is horizontal we don't need the bumper force to keep the car there. I can put the car anywhere that I want to and it will stay. So we just have two forces. The normal force of the plane up, the weight forced down. Those two forces are equal and opposite. Now I'm going to tilt the plane up more and more and more to get up to a vertical angle for the plane or a 90 degree angle. And you can see when I approach 90 that that plane is not exerting any force on the car anymore. All the force is being exerted by the bumper. There's a bumper force up and then the weight force down. And so now those two forces are equal and opposite to each other. Well for any angle in between there will be both a bumper force and a plane force, a normal force from the plane. But those two forces will depend upon the angle. So we're going to see in the segment that follows exactly how those forces depend upon the angle of the plane. We'll begin like always by representing the object, the truck, by point. There are three forces acting on the truck. The weight of the truck which acts vertically downward. That's MG. The normal force which acts perpendicular to the plane and away from it. That's N. And the force of the bumper which acts parallel to the plane and upward along the plane. We'll call that F. We need to set up X and Y axes. It's convenient for that purpose to set up an X axis parallel to the plane. We'll have plus X pointed upward along the plane. And a Y axis perpendicular to the plane will have the plus Y axis pointed upward away from the plane. Now we need to look at the components of the weight force along the X and Y axes. Those will be along the Y axis, this component, along the X axis, this component. To determine what those components are we need to know this angle. And in order to obtain that angle we can relate it to the angle that the plane makes with the horizontal. Call that theta. In order to make the relationship I'm going to extend a couple of lines. I'll extend the Y axis until it crosses the ground. I'll extend the weight force until it does the same thing. Now let's look at a couple of angles. The two angles are the angle of the plane with the horizontal and the angle of the weight force with the Y axis. Notice that both sides of those two angles are perpendicular to each other. The weight force is perpendicular to the ground and the Y axis is perpendicular to the plane. There's a theorem from Geometry that states if two angles have their corresponding sides perpendicular those angles are either equal or supplementary. In this case we can see that they are equal. So that means that the angle we're interested in is the same as the angle theta. Knowing that we can then write that the component of the weight force along the Y axis is the side adjacent to our angle theta and therefore it is mg cosine theta. And the component of mg down the plane is the side opposite the angle theta and that is mg sine theta. Now both of those components are in the negative direction. mg cosine theta is in the negative Y direction so I'll put a minus sign in front of it. mg sine theta is in the negative X direction so I'll put a negative in front of it. Now we're ready to write the net force equations. The net force along the X axis is composed of two forces F which is positive and minus mg sine theta. And the net force along the Y axis also composed of two forces and those are N which is a positive force and mg cosine theta which is negative. So that gives us two net force equations that we can use to complete the analysis of this situation. Now that you've seen the equations for the forces we're going to take some measurements so that you can do some calculations. In order to do that I'm going to replace the normal force of the plane and the bumper force with two other forces. We're going to replace the force of the bumper with the tension force exerted by this string and I'll do that by putting the string over the pulley here and hanging a weight on the end of the string. I'm not going to do that now. I just wanted you to see it and notice that the string is pretty nearly parallel to the plane and I do that because the bumper force is also acting parallel to the plane. Now in order to replace the normal force that force is perpendicular to the plane this string will replace the normal force of the plane and notice that it's perpendicular to the plane just like the normal force would be and I'll drape it over a pulley and I'll put a weight on this side. Now if we knew the mass of the truck then we could calculate what those two weights on the side had to be hanging over the pulleys in order to replace the normal force of the plane and the bumper force. Now I know what the mass is and so I already know what those weights are. I'm going to have you work backwards though. I'm going to put the weights on there. I'm going to tell you what they are and then your job will be to use the equations for the forces to calculate what the mass of the truck is. So I'm going to put those on right now. This particular mass is a total of .270 kilograms. Now that's a mass and we're interested in a weight because that's a force and that is what provides us the tension force in the string here. So you'll need to convert that mass into a weight using the acceleration due to gravity. On this side there's a total mass of .465 kilograms and when I hang that on there notice that the truck moves ever so slightly. You can see that the planes are not needed anymore to support it. I'm going to pull them away and also the bumper is not needed either and so the truck hangs there in the same orientation as if it were on the plane but we've simply replaced the bumper force with the tension force of this string and the normal force of the plane with the tension force of this string. Your job now is to take the two measurements that I've given you and the equations that you have and calculate the mass of the truck and then you'll come back and take another look at the video and we'll measure the mass on a balance and compare to what you came up with.