Chapter 7

Work and Energy Transfer

Section 7.1- Systems and Environments

• System- small portion of the universe being studied– Can be a single object– Can be a collection of objects

• Environment- everything outside the boundaries (physical or not) of the system

• We will generally discuss the conservation of energy of systems rather than individual particles.

7.2- Work

• Energy- the ability to do work– Work is a scalar quantity• The product of Force and Displacment

– Work is only done by forces parallel to the displacement– If F | Δr, no work is done– At any other angle, only the parallel component of the force

does work

cosrFW

rF W

7.2

• Work/Energy have Dimensions of ML2T-2, units = N.m = Joules

• Work is a form of Energy Transfer– Work done on a system (+)– Work done by a system (-)

• Another way of putting it– Energy transferred to the system (+)– Energy transferred from the system (-)

7.2

7.2

• Quick Quizzes p. 185• See Example 7.1 p. 186

7.3 The Scalar Product

• Work is a scalar that results from the multiplication of 2 vectors.– This is known as…• Scalar Product• Dot Product

• Dot Product

– θ is the angle between A and BcosABBA

7.3

• Bcosθ is the projection of B onto A

7.3

• Work is the dot (scalar) product of the Force vector and displacement vector.

cosrF rF

7.3

• Dot Product Properties-– Dot products are commutative

A . B = B . A

– Dot products obey distributive laws of mulitplication

A . ( B + C ) = A . B + A . C

7.3

– If A is | B then A.B = 0 (cos 90)– If A || B then A.B = AB– If A anti-|| B then A.B = -AB

• In Unit vector Notation

0kjkiji

1kkjjii

7.3

• Dealing with Unit Vector Coefficients

Prove this in HW #6

Quick Quiz p. 187Example 7.2, 7.3

zzyyxx

zyx

zyx

BABABABA

kBjBiBB

kAjAiAA

7.4 Work and Varying Force

• is only valid when F is constant.• For a varying F we need to look at very small intervals of Δx(The smaller the interval, the closer Fx becomes to a constant value)

rF W

f

i

x

xx

xxFW

0lim

7.4

• When Δx is infinitely small, the limit of the sum becomes…

• Work is Area under an F vs. x curve.

dxFWf

i

x

x x

7.4

7.4

• Work Done by multiple forces– The Net work done on an object is equal to the

work done by the net force.

– It can also be found by the sum of the work done by all of the individual forces.

f

i

x

net xxW W F dx

net by individual forcesW W

7.4

• Common Application- Work done by a spring– (Hooke’s Law)– x is the position of the attached mass relative to

equilibrium– k is the spring constant (stiffness)– F is always in the opposite direction of x

kxFs

7.4

7.4

• Work Done by the Spring– Calculate the work done on the block by the

spring in moving from xi = xmax to xf = 0

f

i

x

x xs dxFW

0

max

)(xs dxkxW

0221

maxxs kxW

2max2

1 kxWs

7.4

• Area under the curve ½ bh ½ xmax Fmax

½ xmaxkxmax

½ kxmax2

Work done on block is positive, the spring force is forward while the block moves forward.

7.4

Quick Quiz p 192Example 7.6

7.5 Kinetic Energy

• Work is a way of transferring Energy to a system.

• Most commonly this energy now “possessed” by the system is energy of motion

• Kinetic Energy- energy associated with the motion of an object

Work-Kinetic Energy Theorem

KEW

7.5

• The Net Work done on an object will equal its change in Kinetic Energy – Derivation (see board)

Quick Quiz p 195Examples 7.7, 7.8

221 mvK

KEW

7.6 The Non-Isolated System

• Non-Isolated System- external forces from the environment

• Isolated System- no external force (Ch 8)• Work-KE Theorem only valid for Non-Isolated

7.6

• Internal Energy– There are times where we know work is done on

an object yet there is no perceivable ΔKE– Book Sliding across a table• Work is done on the table• The table has no change in Kinetic Energy• Where did that energy go?

– The tables temperature increases (due to the work done on it) we call that Eint

7.6

• Methods of Energy Transfer– Work– Mechanical Waves (ex: sound)– Heat (increase in average particle KE)– Matter Transfer (fuel/convection)– Electrical Transmission (charge passing through

conductor)– Electromagnetic Radiation (Light/UV/IR/radio etc)

7.6

• Energy cannot be created nor destroyed, it is conserved– It can cross the boundary of our system, but it still

exists in the surrounding environment• Quick Quizzes p 199

7.7 Involving Kinetic Friction

• In the case of the book sliding to a stop on the table.– The work done ON the book BY friction is

responsible for the change of kinetic energy to internal energy.

– Or with other forces acting on the object

dFKE k

WdFKE k

7.7

– Or when looking at the book/table system, because there are no outside interactions

– Therefore the result of a friction force is to transform kinetic energy into an equivalent amount of internal energy

0int EKEEsystem

dFE k int

7.7

• Quick Quiz p. 201• Ex 7.9, 7.11

7.8 Power

• While similar tasks often require the same amount of work, they may not take the same time.

• Power- the rate of energy transfer– The rate at which work is done– Refrigerator Example

t

EP

7.8

And so…

Power is the time rate of change of energy/work(derivative)

vF

t

rF

t

W

t

EP

dt

dE

dt

dWP

7.8

• Power has dimensions of ML2T-3 – Units are J/s or Watt– Horsepower 1 hp = 746 W

• Energy described in kWh the energy used for 1 hour at a transfer rate of 1000 W (1 kW, 1000 J/s)

1 kWh = 1000 J/s x 3600 s = 3.6x106 J

7.8

• Quick Quiz p. 204• Example 7.12

7.9 Energy and Automobiles

• Modern Internal Combustion engines are very inefficient using less that 15% of the chemical energy stored in gasoline to power the car. ~ 67% Lost to heat/sound/emr in the engine~ 10% Lost in friction of the drivetrain~ 4 -10% lost to power Fuel Pumps/Alternator/AC

Leaves around 13-19% for Kinetic Energy.

7.9

• When traveling at constant speeds, the total work done is zero (no change in kinetic energy)

• The work done by the engine is dissipated by resistive forces– Rolling friction– Air Resistance ~ v2 (Drag)

art RFF

221 AvDRa

nrr FF

7.9

• Since drag ~ v2 it is the dominant resistance at

high speeds• Rolling Friction is dominant at low speeds.

7.9

• Examples p. 207