# Chapter 5 Work and Energy

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1 Chapter 5 Work and Energy

2 5.1 Work Done by a Constant Force Definition of work: The work done by a constant force acting on an object is equal to the product of the magnitudes of the displacement and the component of the force parallel to that displacement.

3 5.1 Work Done by a Constant Force In (a), there is a force but no displacement: no work is done. In (b), the force is parallel to the displacement.

4 5.1 Work Done by a Constant Force W = F cos(θ) d Unit of work: newton meter (N m) 1 N m is called 1 joule.

5 5.1 Work Done by a Constant Force What if the force that is applied is in the opposite direction of the displacement? What is the best example of such a force?

6 5.1 Work Done by a Constant Force Friction is a force which always opposes the direction of motion, therefore its work done will always be negative. In the case of lifting an object, gravity works against the motion and therefore does negative work.

7 5.2 Work Done by a Variable Force The force exerted by a spring varies linearly with the displacement:

8 5.2 Work Done by a Variable Force A plot of force versus displacement allows us to calculate the work done:

9 Work Done by a Variable Force F s = -kx (ideal spring force) W = ½kx 2 (work done stretching or compressing a spring)

10 What is ENERGY?!

11 5.3 The Work Energy Theorem: Kinetic Energy Kinetic energy is defined: The net work on an object changes its kinetic energy.

12 5.3 The Work Energy Theorem: Kinetic Energy This relationship is called the work energy theorem.

13 Kinetic Energy

14 5.4 Potential Energy Gravitational potential energy: U = m x g x y

15 5.4 Potential Energy Only changes in potential energy are physically significant; therefore, the point where U = 0 may be chosen for convenience.

16 5.4 Potential Energy Potential energy may be thought of as stored work, such as in a compressed spring or an object at some height above the ground. Work done also changes the potential energy (U) of an object.

17 5.4 Potential Energy We can, therefore, define the potential energy of a spring; note that, as the displacement is squared, this expression is applicable for both compressed and stretched springs.

18 Where does the Energy Go? In a perfect system: When you lose potential energy, you gain kinetic energy Example: Object falls what happens? It speeds up! (increased v increased KE) It loses height (decreased y decreased U) When you gain potential energy it is because you are losing kinetic energy Example: Throw an object upwards what happens? It slows down! (decreased v decreased KE) It gains height! (increased y increased U)

19 Conservation (briefly) This balance between kinetic and potential energy is considered conservative because the total energy does not ever change. E Total = KE + U ΔKE + ΔU = 0 When U goes down, KE goes up When KE goes down, U goes up So by extension: ΔU = -ΔKE True story.

20 Conservation (briefly) And in a not-perfect system? Heat Friction Air Resistance Sound Light But we can just forget that stuff right???

21 Energy Equation Recap Work/Energy W = ΔE W = ΔKE W = ΔU Kinetic Energy (KE) KE = ½ mv 2 Potential Energy (U) Gravitational U = mgy ΔU = mgδy Spring U = ½ kx 2 Total Energy (E) E = KE + U ΔKE + ΔU = 0

22 5.5 Conservation of Energy We observe that, once all forms of energy are accounted for, the total energy of an isolated system does not change. This is the law of conservation of energy: The total energy of an isolated system is always conserved. We define a conservative force: A force is said to be conservative if the work done by it in moving an object is independent of the object s path.

23 5.5 Conservation of Energy So, what types of forces are conservative? Gravity is one; the work done by gravity depends only on the difference between the initial and final height, and not on the path between them. Similarly, a nonconservative force: A force is said to be nonconservative if the work done by it in moving an object does depend on the object s path. The quintessential nonconservative force is friction.

24 5.5 Conservation of Energy Another way of describing a conservative force: A force is conservative if the work done by it in moving an object through a round trip is zero. We define the total mechanical energy:

25 5.5 Conservation of Energy For a conservative force: Many kinematics problems are much easier to solve using energy conservation.

26 5.5 Conservation of Energy All three of these balls have the same initial kinetic energy; as the change in potential energy is also the same for all three, their speeds just before they hit the bottom are the same as well.

27 5.5 Conservation of Energy In a conservative system, the total mechanical energy does not change, but the split between kinetic and potential energy does.

28 5.5 Conservation of Energy If a nonconservative force or forces are present, the work done by the net nonconservative force is equal to the change in the total mechanical energy.

29 What is Power?

30 5.6 Power The average power is the total amount of work done divided by the time taken to do the work. If the force is constant and parallel to the displacement,

31 What is the unit for power?

32 5.6 Power The unit for power, J/s, is more commonly referred to as a Watt (W), named for James Watt whose studies in work, energy, and power helped pave the way for modern machine engines. The British unit, horsepower (hp) is a larger unit still commonly used today: 1 hp = 550 ft lb/s = 746 W

33

34 Practice! 1. Fred pushes boulders for a living. If he applied 450 N of force, moving the boulder 4.0 meters in three minutes How much work has he accomplished? What is Fred s power output 2. Bob the horse outputs 3.73x10 3 J of energy over a 5.00 s period. What is his power output in Watts, what is it in HP?

35 5.6 Power Mechanical efficiency: The efficiency of any real system is always less than 100%.

36 5.6 Power

37 5.6 Equation Recap 1 hp = 746 W

38 Practice! A roller coaster lift system is able to lift a 1600 kg rollercoaster up a height of 34.0 m height in 18.0 seconds What is the power output of this motor? If it took 1.78x10 6 J of energy to accomplish this task, what is the efficiency of the lift system?

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