Conservation of Energy Chapter 8 Conservation of Energy 7.3 work done by a varying force 7.4 kinetic Energy and work-energy principle 8.1 Conservative forces 8.2 Potential Energy Chapter Opener. Caption: A polevaulter running toward the high bar has kinetic energy. When he plants the pole and puts his weight on it, his kinetic energy gets transformed: first into elastic potential energy of the bent pole and then into gravitational potential energy as his body rises. As he crosses the bar, the pole is straight and has given up all its elastic potential energy to the athlete’s gravitational potential energy. Nearly all his kinetic energy has disappeared, also becoming gravitational potential energy of his body at the great height of the bar (world record over 6 m), which is exactly what he wants. In these, and all other energy transformations that continually take place in the world, the total energy is always conserved. Indeed, the conservation of energy is one of the greatest laws of physics, and finds applications in a wide range of other fields.

Work done by a constant force Work is an energy transfer that occurs when a force acts on an object that moves. F q Units of work: Nm or Joules (J) Fcos r •Work is done only when force is exerted over a distance. (no displacement=no work) Talk about the car’s motion stopping and forms of energy changing •Only the part of the force parallel to the displacement does work. http://i.telegraph.co.uk/telegraph/multimedia/archive/01435/bmw_1435680c.jpg

Work can be positive, negative or zero F q r N Fpush Work done by WorkGravity = WorkNormal = WorkFriction = WorkFPUSH= Fy Fx q x f mg -fx Negative since f is opposite x Fxcos= Fxx Positive since Fx same direction as x Work done sliding a box across a floor

7-3 Work Done by a Varying Force For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up. Figure 7-9a. Caption: Work done by a force F is (a) approximately equal to the sum of the areas of the rectangles.

7-3 Work Done by a Varying Force In the limit that the pieces become infinitesimally narrow, the work is the area under the curve: Or: Figure 7-9b. Caption: Work done by a force F is (b) exactly equal to the area under the curve of F cos θ vs. l.

7-3 Work Done by a Varying Force Work done by a spring force: You are exerting a force Fp= kx K is the spring constant or stifness. The force exerted by a spring is given by: called restoring force Figure 7-10. Caption: (a) Spring in normal (unstretched) position. (b) Spring is stretched by a person exerting a force FP to the right (positive direction). The spring pulls back with a force FS where FS = -kx. (c) Person compresses the spring (x < 0) and the spring pushes back with a force FS = kx where FS > 0 because x < 0. This is called Hooke’s law

7-3 Work Done by a Varying Force Plot of F vs. x. Work done is equal to the shaded area. Figure 7-11. Caption: Work done to stretch a spring a distance x equals the triangular area under the curve F = kx. The area of a triangle is ½ x base x altitude, so W = ½(x)(kx) = ½ kx2.

7-3 Work Done by a Varying Force Example 7-5: Work done on a spring. (a) A person pulls on a spring, stretching it 3.0 cm, which requires a maximum force of 75 N. How much work does the person do? (b) If, instead, the person compresses the spring 3.0 cm, how much work does the person do? Solution: First calculate the spring constant: k = 2.5 x 103 N/m. Then W = 1.1 J, whether stretching or compressing.

7-3 Work Done by a Varying Force Example 7-6: Force as a function of x. A robot arm that controls the position of a video camera in an automated surveillance system is manipulated by a motor that exerts a force on the arm. The force is given by Figure 7-12. Caption: Robot arm positions a video camera. Solution: Do the integral. W = 0.36 J. where F0 = 2.0 N, x0 = 0.0070 m, and x is the position of the end of the arm. If the arm moves from x1 = 0.010 m to x2 = 0.050 m, how much work did the motor do?

7-4 Kinetic Energy and the Work-Energy Principle Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

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: Figure 7-14. Caption: A constant net force Fnet accelerates a car from speed v1 to speed v2 over a displacement d. The net work done is Wnet = Fnetd.

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: Figure 7-15. Caption: A moving hammer strikes a nail and comes to rest. The hammer exerts a force F on the nail; the nail exerts a force -F on the hammer (Newton’s third law). The work done on the nail by the hammer is positive (Wn = Fd >0). The work done on the hammer by the nail is negative (Wh = -Fd).

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? K = 45 J; this is also the net work.

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? Figure 7-16. The net work is the increase in kinetic energy, 2.5 x 105 J. The net work is the increase in kinetic energy, 2.5 x 105 J.

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). Figure 7-18. Solution: For (a), use the work needed to compress a spring (already calculated). For (b) and (c), use the work-energy principle. W = 2.18 J. 1.54 m/s The drag force does -0.77 J of work; the speed is 1.23 m/s

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. Figure 8-1. Caption: Object of mass m: (a) falls a height h vertically; (b) is raised along an arbitrary two-dimensional path. W=-mg (y2-y1)

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. Figure 8-2. Caption: (a) A tiny object moves between points 1 and 2 via two different paths, A and B. (b) The object makes a round trip, via path A from point 1 to point 2 and via path B back to point 1. (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 nonconservative force. W = FPd Figure 8-3. Caption: A crate is pushed at constant speed across a rough floor from position 1 to position 2 via two paths, one straight and one curved. The pushing force FP is always in the direction of motion. (The friction force opposes the motion.) Hence for a constant magnitude pushing force, the work it does is W = FPd, so if d is greater (as for the curved path), then W is greater. The work done does not depend only on points 1 and 2; it also depends on the path taken.

8-1 Conservative and Nonconservative Forces

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 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: Figure 8-4. Caption: A person exerts an upward force Fext = mg to lift a brick from y1 to y2 . .

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 Example 8-1: 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. Figure 8-5. Answer: a. At point 2, U = 9.8 x 104 J; at point 3, U = -1.5 x 105 J. b. U = -2.5 x 105 J. c. At point 1, U = 1.5 x 105 J. At point 2, U = 2.5 x 105 J. At point 3, U = 0 (by definition); the change in going from point 2 to point 3 is -2.5 x 105 J.

8-2 Potential Energy General definition of gravitational potential energy: For any conservative force: