4.5 Conservative Forces

Transcription

1 4 CONSERVATION LAWS 4.5 Conservative Forces Name: 4.5 Conservative Forces In the last activity, you looked at the case of a block sliding down a curved plane, and determined the work done by gravity as the block descends a vertical distance h. Let s assume that we have a block sliding down a straight plane, instead, as shown to the right. 1. Does the amount of work that gravity does depend on the horizontal distance the block moves? Explain why or why not. 2. Does the amount of work that gravity does depend on whether or not there is friction between the block and the plane? Explain. 3. If there is friction, does the amount of work that friction does depend on only the vertical displacement, or does it also depend on the horizontal displacement? You push the block from the bottom of the plane all the way up to the top of the plane, then let it slide back down. 4. While you are pushing the block up the plane, is the work done by gravity positive or negative? 163

2 4.5 Conservative Forces 4 CONSERVATION LAWS 5. While you are pushing the block up the plane, is the work done by friction positive or negative? 6. As the block slides back down the plane, is the work done by gravity positive or negative? 7. As the block slides back down the plane, is the work done by friction positive or negative? 8. What is the net work done by gravity over the entire cycle (being pushed all the way up and then sliding all the way back down)? Is it positive, negative, or zero? 9. What is the net work done by friction over the entire cycle (being pushed all the way up and then sliding all the way back down)? Is it positive, negative, or zero? 10. A ball is thrown off a cliff at 30 m/s and at some unknown angle. If the cliff is 20 meters high, what is the ball s speed when it lands? 164

3 4 CONSERVATION LAWS 4.5 Conservative Forces 11. Why doesn t the initial angle matter? As you found in the previous problems, the amount of work that gravity does only depends on the vertical displacement, h (or whatever variable we want to use to describe the vertical displacement). This is because the force of gravity here on earth always points downward. Since the amount of work a force does depends only on the portion of the displacement that is parallel to the force (alternatively, we could say it depends only on the portion of the force that is parallel to the displacement), the amount of work gravity does does not depend on how much horizontal displacement an object undergoes. Because of this, we can say that the amount of work done by gravity is path independent. This means that the amount of work done does not depend on the exact path taken to get from one point to another point, it only depends on where the object starts and where it ends. In the case of gravity, it doesn t even depend on the horizontal displacement, either. It only depends on the net vertical displacement between the two points. For example, consider the case of two blocks sliding down differ inclined planes. One plane is straight, the other is curved, as shown below. 12. Assuming a block of mass m on both planes, how much work does gravity do as the blocks slide from an initial elevation of h i to a final elevation of h f? Also explain why gravity does the same amount of work in both cases. 165

4 4.5 Conservative Forces 4 CONSERVATION LAWS Previously, you looked at the amount of work done by both gravity and friction as a block is pushed up an incline and then allowed to slide back down the incline. 13. How much work did gravity do over the full-cycle? How much work did friction do? (In each case, is it a positive, negative, or zero amount of work?) 14. Let s explicitly calculate the amount of work done by friction in this scenario. Assume you have a 20 kg block on a 30 inclined plane with coefficient of friction 0.1. The height of the plane is 1 meter. Calculate the work that friction does during the trip up the plane, and then the trip back down the plane. 166

5 4 CONSERVATION LAWS 4.5 Conservative Forces 15. Let s say you repeatedly push the block up the incline and allow it to slide back down over and over 20 times. After those 20 times, the block is back where it started. How much net work has friction done over those 20 trips? 16. Calculate explicitly the work done by gravity for (a) a trip up and (b) a trip down the plane. 17. How much net work has gravity done over those 20 trips? 167

6 4.5 Conservative Forces 4 CONSERVATION LAWS 18. Keeping in mind why we can say that the work done by gravity is path independent can we say the same thing about the work done by friction? Why or why not? 168