Potential Energy and Conservation of Energy Chapter 8 Potential Energy and Conservation of Energy Copyright © 2014 John Wiley & Sons, Inc. All rights reserved. 1

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Potential energy (symbol = U) is energy (Joules!) Associated with configuration of a system of objects that exert forces on one another A system of objects may be: Earth and a bungee jumper Gravitational potential energy accounts for kinetic energy increase during the fall (KE increases!) Elastic potential energy accounts for deceleration by the bungee cord (KE decreases) © 2014 John Wiley & Sons, Inc. All rights reserved. 2

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Potential energy U is energy that can be associated with the configuration of a system of objects that exert forces on one another Configuration means that WHERE objects are will matter. Physics determines how potential energy is calculated © 2014 John Wiley & Sons, Inc. All rights reserved. 3

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Potential energy U is energy that can be associated with the configuration of a system of objects that exert forces on one another But note: Energy (in the universe) is conserved! Energy can be transformed from potential to kinetic… Energy can be transformed from kinetic to potential…. Energy can be transformed into thermal © 2014 John Wiley & Sons, Inc. All rights reserved. 4

© 2014 John Wiley & Sons, Inc. All rights reserved. Gravitational Potential Energy For an object being raised or lowered: The change in gravitational potential energy is the negative of the work done by the force of gravity © 2014 John Wiley & Sons, Inc. All rights reserved. 5

Gravitational potential energy Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: Force ● Distance Force = mg Distance = Dy Angle between: 0 Work done = POSITIVE Work = +mg Dy

Gravitational potential energy Gravitational potential energy is related to configuration of objects (mass “m” and Earth) Define potential energy of a position at height “h” relative to “0” as mgh NOTE – where “0” is will be YOUR choice…

Gravitational potential energy Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: +mgDy Initial Potential Energy: mgy1 Final Potential Energy: mgy2

Moving DOWN Change in gravitational potential energy is related to work done by gravity. Work done by Gravity: +mgDy (positive!) Initial Potential Energy: mgy1 (start high!) Final PE: mgy2 (lower!) Difference: PEfinal – PE initial (mgy2 – mgy1) < 0 Negative!

Gravitational potential energy Change in gravitational potential energy is related to work done by gravity.. OK… what about moving UPWARDS??

Gravitational potential energy Change in gravitational potential energy is related to work done by gravity.. Work done by Gravity: - mgDy

Gravitational potential energy Change in gravitational potential energy is related to work done by gravity.. Work done by Gravity (up!): - mgDy (negative!) Initial Potential Energy: mgy1 (lower) Final PE: mgy2 (higher!) Difference: PEfinal – PE initial (mgy2 – mgy1) > 0 Positive!

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: - mgDy Difference: PEfinal – PE initial DU = (mgy2 – mgy1) > 0 DU = positive!

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: + mgDy Difference: PEfinal – PE initial DU = (mgy2 – mgy1) < 0 DU = negative!

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy For an object being raised or lowered: © 2014 John Wiley & Sons, Inc. All rights reserved. 15

8-1 Potential Energy Key points: The system consists of two objects A force acts between a particle & rest of system When configuration changes from 1 to 2, force does work W1-2, changing KE to another form When configuration is reversed back (2 to 1), force reverses the energy transfer, doing work W2-1 Thus KE of tomato becomes potential energy, & then kinetic energy again 16

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Conservative forces are forces for which W1-2 = - W2-1 is always true (no matter how the configuration changes and reversals happen!) Examples: gravitational force, spring force © 2014 John Wiley & Sons, Inc. All rights reserved. 17

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Conservative forces are forces for which W1 = -W2 is always true Nonconservative forces are those for which it is false Examples: kinetic friction force, drag force Kinetic energy of a moving particle is transferred to heat by friction Thermal energy cannot be recovered back into kinetic energy of object via friction Therefore the friction is not conservative, thermal energy is not a potential energy © 2014 John Wiley & Sons, Inc. All rights reserved. 18

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy When only conservative forces act on a particle, we find many problems can be simplified: And… Figure 8-4 © 2014 John Wiley & Sons, Inc. All rights reserved. 19

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Figure 8-4 © 2014 John Wiley & Sons, Inc. All rights reserved. 20

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Figure 8-4 © 2014 John Wiley & Sons, Inc. All rights reserved. 21

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.

The conservation of mechanical energy When only force of gravity does work on a system, total mechanical energy of that system is conserved. Gravity is known as a “conservative” force

An example using energy conservation 0.145 kg baseball thrown straight up @ 20m/s. How high?

An example using energy conservation 0.145 kg baseball thrown straight up @ 20m/s. How high? Use Energy Bar Graphs to track total, KE, and PE:

When forces other than gravity do work Now add the launch force! Move hand .50 m upward while accelerating the ball

When forces other than gravity do work Now add the launch force! Move hand .50 m upward while accelerating the ball

Work and energy along a curved path Use same expression for gravitational PE whether path is curved or straight.

Energy in projectile motion 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)

Motion in a vertical circle with no friction Speed at bottom of ramp of radius R = 3.00 m?

Motion in a vertical circle with no friction Speed at bottom of ramp of radius R = 3.00 m?

Motion in a vertical circle with no friction Normal force DOES NO WORK!

Motion in a vertical circle with friction Revisit the same ramp, but this time with friction. If his speed at bottom is 6.00 m/s, what was work by friction?

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 fk? How fast is it moving at the bottom?

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 fk? How fast is it moving at the bottom?

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy For an object being raised or lowered: This also applies to an elastic block-spring system © 2014 John Wiley & Sons, Inc. All rights reserved. 36

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Mathematically: This result allows you to substitute a simpler path for a more complex one if only conservative forces are involved Figure 8-5 © 2014 John Wiley & Sons, Inc. All rights reserved. 37

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy © 2014 John Wiley & Sons, Inc. All rights reserved. 38

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Answer: No. The paths from a → b have different signs. One pair of paths allows the formation of a zero-work loop. The other does not. © 2014 John Wiley & Sons, Inc. All rights reserved. 39

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy For the general case, we calculate work as: So we calculate potential energy as: © 2014 John Wiley & Sons, Inc. All rights reserved. 40

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy So we calculate potential energy as: Using this to calculate gravitational PE, relative to a reference configuration with reference point yi = 0: © 2014 John Wiley & Sons, Inc. All rights reserved. 41

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Use the same process to calculate spring PE: With reference point xi = 0 for a relaxed spring: © 2014 John Wiley & Sons, Inc. All rights reserved. 42

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. Elastic potential energy stored in an ideal spring is Uel = 1/2 kx2. Figure shows graph of elastic potential energy for ideal spring.

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?

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?

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy © 2014 John Wiley & Sons, Inc. All rights reserved. 46

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-1 Potential Energy Answer: (3), (1), (2); a positive force does positive work, decreasing the PE; a negative force (e.g., 3) does negative work, increasing the PE © 2014 John Wiley & Sons, Inc. All rights reserved. 47

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.

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-2 Conservation of Mechanical Energy The mechanical energy of a system is the sum of its potential energy U and kinetic energy K: Work done by conservative forces increases K and decreases U by that amount, so: Using subscripts to refer to different instants of time: In other words: © 2014 John Wiley & Sons, Inc. All rights reserved. 49

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-2 Conservation of Mechanical Energy This is the principle of the conservation of mechanical energy: This is very powerful tool: One application: Choose the lowest point in the system as U = 0 Then at the highest point U = max, and K = min Eq. (8-18) © 2014 John Wiley & Sons, Inc. All rights reserved. 50

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-2 Conservation of Mechanical Energy © 2014 John Wiley & Sons, Inc. All rights reserved. 51

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-2 Conservation of Mechanical Energy Answer: Since there are no nonconservative forces, all of the difference in potential energy must go to kinetic energy. Therefore all are equal in (a). Because of this fact, they are also all equal in (b). © 2014 John Wiley & Sons, Inc. All rights reserved. 52

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.

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.

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 mk = .200

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?

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 + Uint = 0.

A system having two potential energies and friction Gravity, a spring, and friction all act on the elevator. 2000 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.

A system having two potential energies and friction What is the spring constant k for the spring so it stops in 2.00 meters?

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

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve For one dimension, force and potential energy are related (by work) as: Therefore we can find the force F(x) from a plot of the potential energy U(x), by taking the derivative (slope) If we write the mechanical energy out: We see how K(x) varies with U(x): Eq. (8-22) Eq. (8-23) Eq. (8-24) © 2014 John Wiley & Sons, Inc. All rights reserved. 61

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

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 Fx = –U/dx and Fy = –U/dy

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve © 2014 John Wiley & Sons, Inc. All rights reserved. 64

Energy diagrams An energy diagram is a graph that shows both the potential-energy function U(x) and the total mechanical energy E.

Force and a graph of its potential-energy function

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve To find K(x) at any place, take the total mechanical energy (constant) and subtract U(x) Places where K = 0 are turning points There, the particle changes direction (K cannot be negative) At equilibrium points, the slope of U(x) is 0 A particle in neutral equilibrium is stationary, with potential energy only, and net force = 0 If displaced to one side slightly, it would remain in its new position Example: a marble on a flat tabletop © 2014 John Wiley & Sons, Inc. All rights reserved. 67

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve A particle in unstable equilibrium is stationary, with potential energy only, and net force = 0 If displaced slightly to one direction, it will feel a force propelling it in that direction Example: a marble balanced on a bowling ball A particle in stable equilibrium is stationary, with potential energy only, and net force = 0 If displaced to one side slightly, it will feel a force returning it to its original position Example: a marble placed at the bottom of a bowl © 2014 John Wiley & Sons, Inc. All rights reserved. 68

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve Plot (a) shows the potential U(x) Plot (b) shows the force F(x) If we draw a horizontal line, (c) or (f) for example, we can see the range of possible positions x < x1 is forbidden for the Emec in (c): the particle does not have the energy to reach those points Figure 8-9 © 2014 John Wiley & Sons, Inc. All rights reserved. 69

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-3 Reading a Potential Energy Curve Answer: (a) CD, AB, BC (b) to the right © 2014 John Wiley & Sons, Inc. All rights reserved. 70

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-4 Work Done on a System by an External Force We can extend work on an object to work on a system: For a system of more than 1 particle, work can change both K and U, or other forms of energy of the system For a frictionless system: Eq. (8-25) Eq. (8-26) Figure 8-12 © 2014 John Wiley & Sons, Inc. All rights reserved. 71

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-4 Work Done on a System by an External Force For a system with friction: The thermal energy comes from the forming and breaking of the welds between the sliding surfaces Eq. (8-31) Eq. (8-33) Figure 8-13 © 2014 John Wiley & Sons, Inc. All rights reserved. 72

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-4 Work Done on a System by an External Force Answer: All trials result in equal thermal energy change. The value of fk is the same in all cases, since μk has only 1 value. © 2014 John Wiley & Sons, Inc. All rights reserved. 73

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-5 Conservation of Energy Energy transferred between systems can always be accounted for The law of conservation of energy concerns The total energy E of a system Which includes mechanical, thermal, and other internal energy Considering only energy transfer through work: Eq. (8-35) © 2014 John Wiley & Sons, Inc. All rights reserved. 74

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-5 Conservation of Energy An isolated system is one for which there can be no external energy transfer Energy transfers may happen internal to the system We can write: Or, for two instants of time: Eq. (8-36) Eq. (8-37) © 2014 John Wiley & Sons, Inc. All rights reserved. 75

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-5 Conservation of Energy External forces can act on a system without doing work: The skater pushes herself away from the wall She turns internal chemical energy in her muscles into kinetic energy Her K change is caused by the force from the wall, but the wall does not provide her any energy Figure 8-15 © 2014 John Wiley & Sons, Inc. All rights reserved. 76

© 2014 John Wiley & Sons, Inc. All rights reserved. 8-5 Conservation of Energy We can expand the definition of power In general, power is the rate at which energy is transferred by a force from one type to another If energy ΔE is transferred in time Δt, the average power is: And the instantaneous power is: Eq. (8-40) Eq. (8-41) © 2014 John Wiley & Sons, Inc. All rights reserved. 77

© 2014 John Wiley & Sons, Inc. All rights reserved. 8 Summary Conservative Forces Net work on a particle over a closed path is 0 Potential Energy Energy associated with the configuration of a system and a conservative force Eq. (8-6) Gravitational Potential Energy Energy associated with Earth + a nearby particle Elastic Potential Energy Energy associated with compression or extension of a spring Eq. (8-9) Eq. (8-11) © 2014 John Wiley & Sons, Inc. All rights reserved. 78

© 2014 John Wiley & Sons, Inc. All rights reserved. 8 Summary Mechanical Energy For only conservative forces within an isolated system, mechanical energy is conserved Potential Energy Curves At turning points a particle reverses direction At equilibrium, slope of U(x) is 0 Eq. (8-12) Eq. (8-22) Work Done on a System by an External Force Without/with friction: Conservation of Energy The total energy can change only by amounts transferred in or out of the system Eq. (8-26) Eq. (8-33) Eq. (8-35) © 2014 John Wiley & Sons, Inc. All rights reserved. 79

© 2014 John Wiley & Sons, Inc. All rights reserved. 8 Summary Power The rate at which a force transfers energy Average power: Instantaneous power: Eq. (8-40) Eq. (8-41) © 2014 John Wiley & Sons, Inc. All rights reserved. 80