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Equilibrium of Forces, Part 1

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We are going to be looking at the forces acting on an object at equilibrium. 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 is also a bumper that keeps the truck from falling back. Now these strings and these pulleys and strings, we are going to find out what they are doing later, but for now we are going to ignore them.

Let's take a close look at the forces acting on the truck. Since we know the truck is at equilibrium, we know that the forces acting on the truck add up to zero. There are three forces. The weight of the truck, downward, pointed toward the center of the earth. The bumper exerts a force parallel to the plane. And the plane exerts a force perpendicular to the plane, that's a normal force. Those three forces add up to zero. Now let's take a closer look at the forces, and how they depend on the angle of the plane. I'm going to use this setup, because it is a little bit smaller and I can move it around. So here's the bumper, 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. So we only have two forces, the weight down and the normal force up. Now I am going to tilt the plane up more and more and more, to a 90 degree angle, and you can see when I approach 90, the plane is not exerted any force on the car anymore. All the force is being exerted by the bumper. There is the weight force down, and the bumper force up, both equal and opposite. Well, for any angle in between, there is both a bumper force and a plane force, a normal force for the plane. We are going to see in the segment that follows how these two forces depend on the angle of the plane.

We are going to begin as always by representing the object, the truck, by a point. There are three forces that act on the truck. The weight of the truck, that acts vertically downwards. The normal force, that acts perpendicular to the plane and away from it. And the force of the bumper that acts along the plane, we will call that F. We need to set up X and Y axes. It's convenient to set an x axis parallel to the plane and the y axis perpendicular and away from the plane. Now we need to look at the components of the weight force along the X and Y axes. To determine those, we need this angle. In order to obtain this angle, we can relate it to the angle that the plane makes with the horizontal, we will call that θ. In order to make the relationship, I am going to extend a couple of lines. I'm going to extend the y-axis until it reaches the ground, and I'll extend the x-axis until it does the same thing. Now let's look at a couple of angle, the angle of the plane with the horizontal and the angle of the weight force with the normal force. These angles are equal due to geometry. Thus we can write that the component of the weight force along the y-axis is adjacent to the angle of the plane, and is mgcos(θ). And the component of mg down the plane is opposite angle θ, and that is mgsin(θ). Now both of those components are in the negative direction, so we can put negatives in front of both. The net force on the x-axis is Fnet = F-mgsin(θ). The net force on the y-axis is Fnet = F-mgcos(θ). That gives us two net force equations that we can use to complete the analysis of this situation.

Now that you have seen the equations of the forces, we are going to take some measurements. In order to do that I am going to replace the normal force and the bumper force with two other forces. I am going to replace the force of the bumper with a tension force exerted by this string. I will do this by putting this string over a pulley and putting a weight at the end of the string. I am not going to do that now, I just wanted you to see it. And notice that the string is nearly parallel to the plane, and I do that because the bumper force is parallel to the plane. Now to replace the normal force, that force is perpendicular to the plane, this string will replace the normal force of the plane, and notice how it is perpendicular to the plane. I will do this by putting it over a pulley and hanging a weight on the end.

Now if we knew the mass of the truck, then we could calculate the weights that the two pulleys needed to be in order to replace the forces of the normal force and bumper force. Now I know what the mass is, so I know what those weights are. I am going to have you work backwards thought, I am going to put those weights on there, and have you use those forces to find what the weight of the truck is. I am going to do that now. This particular mass is a total of 0.270kg, now that's a mass, and we are interested in a weight, because that force transfers to the tension force. So you will need to convert that mass into a weight. This mass is about 0.465 kg. The truck moves ever so slightly, and the plane is not needed to hold the truck anymore. And also the bumper is not needed either. The truck hangs there in the same orientation if the plane was still there. Your job now is to take the measurements I have given you and calculate the weight of the truck.



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