Work example problems

I'm going to show you some examples of how to solve problems involving work. Imagine a 4 kilogram trashcan. The trashcan is disgusting. So someone ties a string to it and pulls on the string with a force of 50 newtons. The force of kinetic friction on the trashcan while it slides is 30 newtons. The trash can slides across the ground for a distance of 10 meters. Let's try to find the work done by each force on the trash can as it slides across the ground. To find the work done by each force, we should recall the formula definition of work. Work equals Fd cosine theta, where theta is the angle between the force doing the work and the direction the trashcan is moving. There are four forces involved here-- tension, the normal force, the gravitational force, and the force of kinetic friction. In finding the work done for all of these forces, the displacement is going to be 10 meters. But the value of the force and the angle between that force and the displacement is going to differ for each of the forces. For instance, to find the work done by the force of tension, we'll plug in the size of the tension, which is 50 newtons. The displacement is 10 meters. And since the tension force is pointed in the same direction as the displacement, the angle between the force of tension and the displacement is 0 degrees. And since cosine of 0 is 1, the work done by the tension force is 500 joules. To find the work done by friction, we'll plug in the size of the force of friction, which is 30 newtons. The displacement is still 10 meters. And since the force of friction points in the opposite direction as the displacement, the angle between the force of friction and the displacement is 180 degrees. Since cosine of 180 is negative 1, the work done by the force of friction is negative 300 joules. Now let's figure out the work done by the gravitational force. The force of gravity is mg. So the force of gravity is 4 kilograms times 9.8 meters per second squared, which is 39.2 newtons. The displacement is again 10 meters. But the angle between the gravitational force and the direction of the displacement is 90 degrees in this case. And since cosine of 90 is 0, the gravitational force does no work on this trashcan. Similarly, if we were to find the work done by the normal force, the angle between the direction of the displacement and the normal force is 90 degrees. So the normal force also does no work on the trashcan. This makes sense because forces that are perpendicular to the motion can never do any work on that object. So that's how you can find the work done by individual forces. And if we wanted to know the net work done on this trashcan, we could just add up the work done by each individual force. So the net work is going to be 200 joules. Now that we know the net work done on the trashcan, we can use the work-energy principle to figure out the speed of the trashcan after it's slid the 10 meters. The work-energy principle says that the net work done on an object is equal to the change in kinetic energy of that object. So 200 joules is going to equal the difference in kinetic energy. If we assume the trashcan started at rest, which seems reasonable, the initial velocity is 0. So we can solve for the final speed of the trashcan, which comes out to be 10 meters per second. This time, let's say you take the trashcan and lift it upwards with a constant velocity for a distance of 2 meters. In order to lift the trashcan up with a constant velocity, you need to push with a force equal to the weight of the trashcan, which means you have to push upwards with a force of 39.2 newtons. So to find the work done by the force that you exert, the force is going to be 39.2 newtons. The displacement is going to be 2 meters. And the angle between the force and the displacement is going to be 0 degrees because the direction of the force that you exert is in the same direction as the displacement of the trashcan. So the work that you've done in lifting up this 4 kilogram trashcan is 78.4 joules. To find the work done by the force of gravity, we can use the force of gravity, which is again 39.2 newtons. The displacement is again 2 meters. But the angle between the direction of the displacement and the gravitational force is 180 degrees because the displacement points up and the gravitational force points down. So the work done by the gravitational force is negative 78.4 joules, which means the net work done on the trashcan is 0. And that makes sense. Because since the trashcan moved upwards with constant velocity, there was no change in the kinetic energy of this object.