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

Copyright © 2014 John Wiley & Sons, Inc. All rights reserved.

Potential Energy and Conservation of Energy

Chapter 8

© 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:o Earth and a bungee jumpero Gravitational potential energy

accounts for kinetic energy increase during the fall (KE increases!)

o Elastic potential energy accounts for deceleration by the bungee cord (KE decreases)



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



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


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

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 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…



Work done by Gravity: +mgDy

Initial Potential Energy:

mgy1

Final Potential Energy:

mgy2

Moving DOWN


Work done by Gravity: +mgDy

(positive!)


mgy1 (start high!)

Final PE:

mgy2 (lower!)

Difference: PEfinal – PE initial

(mgy2 – mgy1) < 0

Negative!


Change in gravitational potential energy is related to work done by gravity..

OK… what about moving

UPWARDS??



Work done by Gravity:

- mgDy



Work done by Gravity (up!):

- mgDy (negative!)


mgy1

(lower)

Final PE:

mgy2

(higher!)


(mgy2 – mgy1) > 0

Positive!


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


DU = (mgy2 – mgy1) > 0

DU = positive!


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


DU = (mgy2 – mgy1) < 0

DU = negative!





Key points:

1. The system consists of two objects

2. A force acts between a particle & rest of system

3. When configuration changes from 1 to 2, force does work W

1-2, changing KE to another form

4. When configuration is reversed back (2 to 1), force reverses the energy transfer, doing work W

2-1

Thus KE of tomato becomes potential energy, & then kinetic energy again



Conservative forces are forces for which

W1-2 = - W2-1

is always true (no matter how the configuration changes and reversals happen!)

o Examples: gravitational force, spring force



Conservative forces are forces for which W1 = -W2 is always true

Nonconservative forces are those for which it is falseo Examples: kinetic friction force, drag forceo Kinetic energy of a moving particle is transferred to heat by

frictiono Thermal energy cannot be recovered back into kinetic

energy of object via frictiono Therefore the friction is not conservative, thermal energy is

not a potential energy


When only conservative forces act on a particle, we find many problems can be simplified:

And…


Figure 8-4



Figure 8-4



Figure 8-4

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?


Speed at bottom of ramp of radius R = 3.00 m?


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?




This also applies to an elastic block-spring system



Mathematically:

This result allows you to substitute a simpler path for a more complex one if only conservative forces are involved

Figure 8-5





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.


For the general case, we calculate work as:

So we calculate potential energy as:



So we calculate potential energy as:

Using this to calculate gravitational PE, relative to a reference configuration with reference point y

i = 0:



Use the same process to calculate spring PE:

With reference point xi = 0 for a relaxed spring:


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?





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

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.


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:

8-2 Conservation of Mechanical Energy



This is the principle of the conservation of mechanical energy:

This is very powerful tool:

One application:o Choose the lowest point in the system as U = 0

o Then at the highest point U = max, and K = min

Eq. (8-18)





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).

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


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)

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



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



To find K(x) at any place, take the total mechanical energy (constant) and subtract U(x)

Places where K = 0 are turning pointso 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 = 0o If displaced to one side slightly, it would remain in its new

position

o Example: a marble on a flat tabletop



A particle in unstable equilibrium is stationary, with potential energy only, and net force = 0o If displaced slightly to one direction, it will feel a force

propelling it in that direction

o Example: a marble balanced on a bowling ball

A particle in stable equilibrium is stationary, with potential energy only, and net force = 0o If displaced to one side slightly, it will feel a force returning it

to its original position

o Example: a marble placed at the bottom of a bowl



Figure 8-9

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



Answer: (a) CD, AB, BC (b) to the right


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



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



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.


8-5 Conservation of Energy

Energy transferred between systems can always be accounted for

The law of conservation of energy concernso The total energy E of a systemo Which includes mechanical, thermal, and other internal

energy

Considering only energy transfer through work:

Eq. (8-35)



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)



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



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. (8-41)

Eq. (8-40)


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

8 Summary

Eq. (8-9)

Gravitational Potential Energy

Energy associated with Earth + a nearby particle

Eq. (8-11)

Eq. (8-6)

Elastic Potential Energy Energy associated with

compression or extension of a spring


Eq. (8-35)

8 Summary

Eq. (8-12)

Work Done on a System by an External Force Without/with friction:

Eq. (8-33)

Conservation of Energy The total energy can change

only by amounts transferred in or out of the system

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. (8-22)

Eq. (8-26)


Power The rate at which a force

transfers energy

Average power:

Instantaneous power:

8 Summary

Eq. (8-40)

Eq. (8-41)

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