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Chapter 8 Conservation of Energy HW5:Chapter 8: Pb.8, Pb.20, Pb.23, Pb.33, Pb.68, Pb.71 Due Wednesday, Oct.19 7-4 Kinetic Energy and the Work-Energy Principle If we write the acceleration in terms of the velocity and the distance, we find that the work done here is We define the kinetic energy as: 7-4 Kinetic Energy and the Work-Energy Principle This means that the work done is equal to the change in the kinetic energy: This is the Work-Energy Principle: • If the net work is positive, the kinetic energy increases. • If the net work is negative, the kinetic energy decreases. 7-4 Kinetic Energy and the Work-Energy Principle Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules. Energy can be considered as the ability to do work: 7-4 Kinetic Energy and the Work-Energy Principle Example 7-7: Kinetic energy and work done on a baseball. A 145-g baseball is thrown so that it acquires a speed of 25 m/s. (a) What is its kinetic energy? (b) What was the net work done on the ball to make it reach this speed, if it started from rest? 7-4 Kinetic Energy and the Work-Energy Principle Example 7-8: Work on a car, to increase its kinetic energy. How much net work is required to accelerate a 1000-kg car from 20 m/s to 30 m/s? The net work is the increase in kinetic energy 7-4 Kinetic Energy and the Work-Energy Principle Example 7-10: A compressed spring. A horizontal spring has spring constant k = 360 N/m. (a) How much work is required to compress it from its uncompressed length (x = 0) to x = 11.0 cm? (b) If a 1.85-kg block is placed against the spring and the spring is released, what will be the speed of the block when it separates from the spring at x = 0? Ignore friction. (c) Repeat part (b) but assume that the block is moving on a table and that some kind of constant drag force FD = 7.0 N is acting to slow it down, such as friction (or perhaps your finger). Problem 56 56. (II) An 85-g arrow is fired from a bow whose string exerts an average force of 105 N on the arrow over a distance of 75 cm. What is the speed of the arrow as it leaves the bow? 8-1 Conservative and Nonconservative Forces 2 1 Example 8-1: How much work is needed to move a particle from position 1 to 2? 8-1 Conservative and Nonconservative Forces A force is conservative if: the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken. Example: gravity. 8-1 Conservative and Nonconservative Forces Another definition of a conservative force: a force is conservative if the net work done by the force on an object moving around any closed path is zero. (a) (b) 8-1 Conservative and Nonconservative Forces If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a non-conservative force. 8-1 Conservative and Nonconservative Forces 8-2 Potential Energy Example 8-2 What potential energy is needed to move a block upward with an external force Fext? 8-2 Potential Energy In raising a mass m to a height h, the work done by the external force is We therefore define the gravitational potential energy at a height y above some reference point: 8-2 Potential Energy Example 8-3: Potential energy changes for a roller coaster. A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1. (b) What is the change in potential energy when the car goes from point 2 to point 3? (c) Repeat parts (a) and (b), but take the reference point (y = 0) to be at point 3. 8-2 Potential Energy An object can have potential energy by virtue of its surroundings. Potential energy can only be defined for conservative forces Familiar examples of potential energy: • A wound-up spring • A stretched elastic band • An object at some height above the ground 8-2 Potential Energy This potential energy can become kinetic energy if the object is dropped. Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces). If Ugrav = mgy, where do we measure y from? It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured. 8-2 Potential Energy General definition of gravitational potential energy: For any conservative force: Problem 7 Problem 9 8-2 Potential Energy A spring has potential energy, called elastic potential energy, when it is compressed. The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring. 8-2 Potential Energy Then the potential energy for a spring is: 8-2 Potential Energy In one dimension, We can invert this equation to find U (x) if we know F (x): In three dimensions: 8-3 Mechanical Energy and Its Conservation If the forces are conservative, the sum of the changes in the kinetic energy and in the potential energy is zero—the kinetic and potential energy changes are equal but opposite in sign. This allows us to define the total mechanical energy: And its conservation: . 8-3 Mechanical Energy and Its Conservation The principle of conservation of mechanical energy: If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant—it is conserved. Problem 16 (II) A 72-kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.5m/s. (a) How fast is he going as he lands on the trampoline, 2.0 m below (b) If the trampoline behaves like a spring of spring constant 5.8 × 104 N/m, how far does he depress it?