Chapter 8 - Potential Energy and Conservation of Energy

• Conservative vs. Non-conservative Forces• Definition of Potential Energy• Conservation Of Mechanical Energy • Determining Potential Energy

– Gravitational – near the surface of the earth– Gravitational – anywhere - escape velocity– Elastic

• Determining the Force From Potential Energy Functions• Work Done by Non-conservative Forces• Power

Power

• Rate at which work is done.

• Average Power

• Instantaneous Power

WPt

dWP F vdt

UnitsPhysicalQuantity

DimensionSymbol

SI MKS SI CGS USCustomary

Length [L] m cm ft

Mass [M] kg g slug

Time [T] sec sec sec

Acceleration [L/T2] m/s2 cm/s2 ft/s2

Force [M-L/T2] newton (N)kg-m/s2

Dyneg-cm/s2

pound (lb)slug- ft/s2

Energy [M-L2/T2] Joule (J)N-m

kg-m2/s2

ErgDyne-cmg-cm2/s2

Ft-lbslug-ft2/s2

Power [M-L2/T3]Watt (W)

J/s N-m/s

HP =550 ft-lb/s

Conservative vs. Non-conservative

• Conservative - A force is said to be conservative if the work done by the force acting on a object moving between two points “is independent of the path” the particle takes between the points.

• Non-conservative - “depends on the path”

b

aW F dr

dr

Example: Gravity near the surface of the earth

ˆF mgj

ˆ ˆ ˆdr dxi dyj dzk

b

aW F dr

2 1W mg y y mgh

dr

Alternative definition

• A force is conservative if the net work done by the force on an object moving around any closed path is zero.

Gravity is a conservative force!

A nonconservative force

Friction is a nonconservative force!

Potential Energy

• Energy associated with the position or configuration of a system.

• The change in potential energy associated with a particular conservative force is the negative of the work done by that force.

b

aW F dr

2 1U U U W

Examples:

• Gravity

• Springs

G 2 1W mg y y 2 1U mg y y

U mgy

2S

1W kx2

21U kx2

21U kx2

Differential form

xU F dx 2

1U F dr

xdUFdx

One dimension:

Three dimensions:

xUFx

yUFy

zUFz

U U Uˆ ˆ ˆF i j k Ux y z

Potential Energy Summary

• Potential energy is only associated with conservative forces. It is the negative of the work done by the conservative force.

• The zero point of potential energy is arbitrary and should be chosen where it is most convienient.

• Potential energy is not something a body has by itself, but rather is associated with the interaction of two or more objects.

Conservation of Mechanical Energy

W K Work-Energy Principle

W U Definition of Potential Energy

0 K U

1 1 2 2K U K U

1 2E E

Problem solving strategy

Who is going faster at the bottom?• Assume no friction• Assume both have the same

speed pushing off at the top

Problem 1• A Block of mass m is released from rest and

slides down a frictionless track of height h above a table. At the bottom of the track, where the surface is horizontal, the block strikes and sticks to a light spring.

• Find the maximum distance the spring is compressed.

• m = 2 kg, h = 1 m, k = 490 N/m

Problem 2• A ball (mass m) on a

string (length L) is released from rest with the string horizontal. What is the speed when it reaches its lowest point?

• What if the string was not horizontal, instead being released from some angle ?

Energy conservation with dissipative forces

• Total energy is neither increased or decreased in any process. Energy can be transformed from one form to another, and transferred from one body to another, but the total amount remains constant.

C NC NCW W W U W K

NCW K U

Example 3• A roller coaster with mass of

1000 kg starts at a height of 40 m and is found to reach a height of only 25 m before coming to a stop. It traveled a distance of 400 m. Estimate the average friction force.

• Is the friction force constant?

Problem 7• A 2 kg block is attached to a light spring of force constant

500 N/m. The block is pulled 5 cm to the right and of equilibrium. How much work is required to move the block?

• If released from rest, find the speed of the block as it passes back through the equilibrium position if– the horizontal surface is frictionless.– the coefficient of friction is 0.35.

Example

• A ball of mass 4.64 kg is taken to a position 3 moon radii above the surface of the moon where it is dropped from rest. What is the speed of the ball as it just starts to make contact with the surface of the moon? – Mm = 7.35 x 1022 kg

– Rm = 1.74 x 106 m

Gravitational potential energy again

E2

Gm m ˆF rr

2

1W F dl

W U

EGM mU rr

Escape velocity

2 Eesc

E

GM m1 mv 02 r

Eesc

E

2GMvr