PHYC 160 Power, Gravitational Potential Energy Lecture #17 Gravitational Work • Let’s examine the gravitational work done on an object when it moves along some arbitrary path: y final WGravity final FGravity ds initial dxiˆ dyjˆ initial final final mgjˆ dxiˆ initial mgjˆ dyjˆ initial final 0 mg x mgjˆ initial mg y dy ˆj ˆj mg y final yiniti Work (by gravity) on a Pendulum ds ds Mg Mg CPS Question 16-2 • How much work does the cable holding the bowling ball do from the top of the arc to the bottom? A) +mgh FT B) -mgh C) 0 D) Not enough information to solve. h FT ds ds Work-Energy Theorem WTOTAL WGravity WElastic WOther KE • Now, we have just broken the total work on an object up into three terms, the first two being from forces that we understand: final WGravity FGravity ds mg y f FElastic ds 1 2 kx 2 yi initial final WElastic initial Sign determined by the dot product (+ when pushing in same direction as motion, - when opposite) Power • Power is the rate at which work is done. • Average power is Pav = ΔW/Δt and instantaneous power is P = dW/dt. • The SI unit of power is the watt (1 W = 1 J/s), but other familiar units are the horsepower and the kilowatt-hour. Copyright © 2012 Pearson Education Inc. Gravitational potential energy • Energy associated with position is called potential energy. • Gravitational potential energy is Ugrav = mgy. • Figure 7.2 at the right shows how the change in gravitational potential energy is related to the work done by gravity. Copyright © 2012 Pearson Education Inc. Q7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases. © 2012 Pearson Education, Inc. The conservation of mechanical energy • • • The total mechanical energy of a system is the sum of its kinetic energy and potential energy. A quantity that always has the same value is called a conserved quantity. When only the force of gravity does work on a system, the total mechanical energy of that system is conserved. This is an example of the conservation of mechanical energy. Figure 7.3 below illustrates this principle. Copyright © 2012 Pearson Education Inc. An example using energy conservation • Refer to Figure 7.4 below as you follow Example 7.1. • Notice that the result does not depend on our choice for the origin. Copyright © 2012 Pearson Education Inc. Q7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y2. What is the speed of the ball when it is at a height of y2/2? Ignore air resistance. A. 10.0 m/s v2 = 0 v1 = 20.0 m/s m = 0.150 kg B. less than 10.0 m/s but greater than zero C. greater than 10.0 m/s D. not enough information given to decide © 2012 Pearson Education, Inc. y2 y1 = 0 When forces other than gravity do work • Refer to ProblemSolving Strategy 7.1. • Follow the solution of Example 7.2. Copyright © 2012 Pearson Education Inc. Work and energy along a curved path • We can use the same expression for gravitational potential energy whether the body’s path is curved or straight. Copyright © 2012 Pearson Education Inc. Energy in projectile motion • Two identical balls leave from the same height with the same speed but at different angles. • Follow Conceptual Example 7.3 using Figure 7.8. Copyright © 2012 Pearson Education Inc. Motion in a vertical circle with no friction • Follow Example 7.4 using Figure 7.9. Copyright © 2012 Pearson Education Inc. Q7.4 The two ramps shown are both frictionless. The heights y1 and y2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the right-hand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track. © 2012 Pearson Education, Inc. Moving a crate on an inclined plane with friction • • Follow Example 7.6 using Figure 7.11 to the right. Notice that mechanical energy was lost due to friction. Copyright © 2012 Pearson Education Inc. Conservative and Non-conservative • The nice thing about the conservation of mechanical energy is that the change in the potentials only are determined by the initial and final points of the path. That’s because potentials always describe conservative forces – forces where the work done by them in going from one point to another is path independent. Non- conservative forces • An example of a non-conservative force is friction. The work done by friction is definitely dependent on the path. • Let’s take the example of moving a book on a table with kinetic friction: Path 2 Path 1 • Since path 2 is longer, there will be more work done by friction. Work done by a spring • Figure 7.13 below shows how a spring does work on a block as it is stretched and compressed. Copyright © 2012 Pearson Education Inc. Work from a Spring • The same thing can be done with the work from a spring: final final WElastic FElastic ds initial kxiˆ dxiˆ initial final kxiˆ dxiˆ initial final k initial xdx iˆ iˆ 1 2 k x final 2 2 initial x Work from a Spring • If we take the initial position to be the relaxed position of the spring, then set xinitial = 0, we have 1 2 WElastic 2 kx final • And we can make the same definition of the elastic potential energy: U Elastic WElastic 1 2 kx 2 • Where, again, this is the energy stored in the spring-block system. Elastic potential energy • A body is elastic if it returns to its original shape after being deformed. • Elastic potential energy is the energy stored in an elastic body, such as a spring. • The elastic potential energy stored in an ideal spring is Uel = 1/2 kx2. • Figure 7.14 at the right shows a graph of the elastic potential energy for an ideal spring. Copyright © 2012 Pearson Education Inc. Q7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy Ugrav and the elastic potential energy Uel? A. Ugrav and Uel are both increasing. B. Ugrav and Uel are both decreasing. C. Ugrav is increasing; Uel is decreasing. D. Ugrav is decreasing; Uel is increasing. E. The answer depends on how the block’s speed is changing. © 2012 Pearson Education, Inc.
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