PHYC 160 Elastic and Gravitational Potential Energy Lecture #18 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 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. 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. Stretching a spring • The force required to stretch a spring a distance x is proportional to x: Fx = kx. • k is the force constant (or spring constant) of the spring. • The area under the graph represents the work done on the spring to stretch it a distance X: W = 1/2 kX2. 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. Situations with both gravitational and elastic forces • When a situation involves both gravitational and elastic forces, the total potential energy is the sum of the gravitational potential energy and the elastic potential energy: U = Ugrav + Uel. • • Figure 7.15 below illustrates such a situation. Follow Problem-Solving Strategy 7.2. 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. Conservative and nonconservative forces • A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative. • The work done between two points by any conservative force a) can be expressed in terms of a potential energy function. b) is reversible. c) is independent of the path between the two points. d) is zero if the starting and ending points are the same. • A force (such as friction) that is not conservative is called a nonconservative force, or a dissipative force. 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. Force and potential energy in one dimension • In one dimension, a conservative force can be obtained from its potential energy function using Fx(x) = –dU(x)/dx • Figure 7.22 at the right illustrates this point for spring and gravitational forces. • Follow Example 7.13 for an electric force. Copyright © 2012 Pearson Education Inc. Energy diagrams • An energy diagram is a graph that shows both the potential-energy function U(x) and the total mechanical energy E. • Figure 7.23 illustrates the energy diagram for a glider attached to a spring on an air track. Copyright © 2012 Pearson Education Inc. Force and a graph of its potential-energy function • Figure 7.24 below helps relate a force to a graph of its corresponding potential-energy function. 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