Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Chapter 7 Potential Energy and Energy Conservation

Copyright © 2012 Pearson Education Inc. Goals for Chapter 7 To use gravitational potential energy for vertical motion To use elastic potential energy for a body attached to a spring To solve problems involving conservative and non-conservative forces

Copyright © 2012 Pearson Education Inc. Potential Energy Energy associated with a particular position of a body when subjected to or acted on by forces. –Field Forces act on bodies even if not touching, like gravity, magnetism, electricity –Direct Contact forces, like springs Energy associated with position is potential energy. –Example: Gravitational potential energy is U grav = mgy for a position “y”. But what is “y”? Where is “y” = 0?? Potential Energy is RELATIVE, not absolute…

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity.

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: +mg  y

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: +mg  y Initial Potential Energy: mgy 1 Final Potential Energy: mgy 2

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: +mg  y Initial Potential Energy: mgy 1 Final PE: mgy 2 Difference: PE final – PE initial (mgy 2 – mgy 1 ) <0

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity..

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Change in gravitational potential energy is related to work done by gravity.. Work done by Gravity: - mg  y Initial Potential Energy: mgy 1 Final PE: mgy 2 Difference: PE final – PE initial (mgy 2 – mgy 1 ) >0

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Either moving DOWN or UP, change in gravitational potential energy is equal in magnitude and opposite in sign to work done by gravity. Work done by Gravity down: mg  y Difference: PE final – PE initial (mgy 2 – mgy 1 ) < 0

Copyright © 2012 Pearson Education Inc. Gravitational potential energy Either moving DOWN or UP, change in gravitational potential energy is equal in magnitude and opposite in sign to work done by gravity. Work done by Gravity up: -mg  y Difference: PE final – PE initial (mgy 2 – mgy 1 ) >0

Copyright © 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.

Copyright © 2012 Pearson Education Inc. The conservation of mechanical energy When only force of gravity does work on a system, total mechanical energy of that system is conserved. This is an example of the conservation of mechanical energy. Gravity is known as a “conservative” force

Copyright © 2012 Pearson Education Inc. An example using energy conservation kg baseball thrown straight 20m/s. How high?

Copyright © 2012 Pearson Education Inc. An example using energy conservation kg baseball thrown straight 20m/s. How high? Use Energy Bar Graphs to track total, KE, and PE:

Copyright © 2012 Pearson Education Inc. When forces other than gravity do work Now add the launch force! (move hand.50 m upward while accelerating the ball)

Copyright © 2012 Pearson Education Inc. When forces other than gravity do work Now add the launch force! (move hand.50 m upward while accelerating the ball)

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 – example 7.3 Two identical balls leave from the same height with the same speed but at different angles. Prove they have the same speed at any height h (neglecting air resistance)

Copyright © 2012 Pearson Education Inc. Motion in a vertical circle with no friction Speed at bottom of ramp of radius R = 3.00 m?

Copyright © 2012 Pearson Education Inc. Motion in a vertical circle with no friction Speed at bottom of ramp of radius R = 3.00 m?

Copyright © 2012 Pearson Education Inc. Motion in a vertical circle with no friction Normal force DOES NO WORK!

Copyright © 2012 Pearson Education Inc. Motion in a vertical circle with friction Revisit the same ramp as in the previous example, but this time with friction. If his speed at bottom is 6.00 m/s, what was work by friction?

Copyright © 2012 Pearson Education Inc. Moving a crate on an inclined plane with friction Slide 12 kg crate up 2.5 m incline without friction at 5.0 m/s. With friction, it goes only 1.6 m up the slope. What is f k ? How fast is it moving at the bottom?

Copyright © 2012 Pearson Education Inc. Moving a crate on an inclined plane with friction Slide 12 kg crate up 2.5 m incline without friction at 5.0 m/s. With friction, it goes only 1.6 m up the slope. What is f k ? How fast is it moving at the bottom?

Copyright © 2012 Pearson Education Inc. Work done by a spring Work on a block as spring is stretched and compressed.

Copyright © 2012 Pearson Education Inc. 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 U el = 1/2 kx 2. 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 = U grav + U el. Figure 7.15 below illustrates such a situation. Follow Problem-Solving Strategy 7.2.

Copyright © 2012 Pearson Education Inc. Motion with elastic potential energy Glider of mass 200 g on frictionless air track, connected to spring with k = 5.00 N/m. Stretch it 10 cm, and release from rest. What is velocity when x = 0.08 m?

Copyright © 2012 Pearson Education Inc. Motion with elastic potential energy Glider of mass 200 g on frictionless air track, connected to spring with k = 5.00 N/m. Stretch it 10 cm, and release from rest. What is velocity when x = 0.08 m?

Copyright © 2012 Pearson Education Inc. A system having two potential energies and friction Gravity, a spring, and friction all act on the elevator kg elevator with broken cables moving at 4.00 m/s Contacts spring at bottom, compressing it 2.00 m. Safety clamp applies constant 17,000 N friction force as it falls.

Copyright © 2012 Pearson Education Inc. A system having two potential energies and friction What is the spring constant k for the spring so it stops in 2.00 meters?

Copyright © 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.

Copyright © 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. A force (such as friction) that is not conservative is called a non-conservative force, or a dissipative force.

Copyright © 2012 Pearson Education Inc. Frictional work depends on the path Move 40.0 kg futon 2.50 m across room; slide it along paths shown. How much work required if  k =.200

Copyright © 2012 Pearson Education Inc. Conservative or nonconservative force? Suppose force F = Cx in the y direction. What is work required in a round trip around square of length L?

Copyright © 2012 Pearson Education Inc. Conservation of energy Nonconservative forces do not store potential energy, but they do change the internal energy of a system. The law of the conservation of energy means that energy is never created or destroyed; it only changes form. This law can be expressed as  K +  U +  U int = 0.

Copyright © 2012 Pearson Education Inc. Force and potential energy in one dimension In one dimension, a conservative force can be obtained from its potential energy function using F x (x) = –dU(x)/dx

Copyright © 2012 Pearson Education Inc. Force and potential energy in one dimension In one dimension, a conservative force can be obtained from its potential energy function using F x (x) = –dU(x)/dx

Copyright © 2012 Pearson Education Inc. Force and potential energy in two dimensions In two dimension, the components of a conservative force can be obtained from its potential energy function using F x = –  U/dxand F y = –  U/dy

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.

Copyright © 2012 Pearson Education Inc. Force and a graph of its potential-energy function