1. Conservation of Energy
Chapter 8
Conservation of Energy
7.4 kinetic Energy and work-energy principle
8.1 Conservative forces
8.2 Potential Energy
8.3 Mechanical Energy and
Its Conservationial Energy
8.4 Problem Solving Using Conservation of Mechanical 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.

2. Question
If you have a variable Force, you can find the work by finding: A) The area under a curve of Force as a function of time B) The area under a curve of Force as a function of displacement C) The slope curve of Force as a function of time D) The slope of a curve of Force as a function of displacement

3. 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

4. 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 J of work; the speed is 1.23 m/s

5. 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?

6. 7-4 Kinetic Energy and the Work-Energy Principle
Energy was traditionally defined as the ability to do work. All forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

7. 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 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.

8. 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.

9. 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 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).

10. 8-1 Conservative and Nonconservative Forces
Example 8-1: How much work is needed to move a particle from position 1 to 2? 2 1

11. 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)

12. 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)

13. 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.
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.

14. 8-1 Conservative and Nonconservative Forces

15. 8-2 Potential Energy
Example 8-2 What potential energy is needed to move a block upward with an external force Fext?

16. 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 . .

17. 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.
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.

18. Problem 7