Work & EnergyWork & Energy

Physics, Chapter 5Physics, Chapter 5

WorkWork

Section 5.1Section 5.1

Definition of WorkDefinition of Work

• In Physics, work means more than something that requires physical or mental effort

• Work is done on an object when a force causes a displacement of the object

• Work is the product of the force applied to an object and the displacement of the object

FdW

Caution!Caution!

• Work is done only when a force or a component

of a force is parallel to a displacement!

Net Work Done by a Constant Net Work Done by a Constant Net ForceNet Force

cosdFW netnet

θ

F

F cos θ

d

Units of Work

• The SI unit of work is the joulejoule (J)• Derived from the formula for work

• The joule is the unit of energy, thus….• Work is a type of energy transfer!!

2

2

s

mkg 1 mN 1 J 1

FdW

Sample Problem ASample Problem A

• How much work is done on a sled pulled 4.00 m to the right by a force of 75.0 N at an angle of 35.0° above the horizontal?

F

θ

J 246

0.35cosm 00.4N 0.75

cos

W

W

FdW d

Fd cosFd cosθθ

F

Fg

Fup

θ

d

How much work was done by Fg on the sled?

How much work was done by Fup on the sled?

If the force of kinetic friction was 20.0 N, how much work was done by friction on the sled?

Fk

FN

Fy Fx

The Sign of WorkThe Sign of Work

• Work is a scalar quantity and can be positive or negative

• Work is positive when the component force & displacement have the same direction

• Work is negative when they have opposite directions

F

Fg

Fup

θ

J 246

0.35cosm 00.4N 0.75

cos

W

W

FdW

d

If the force of kinetic friction was 20.0 N, how much work was done by friction on the sled?

Wf = Fk∙d = (-20.0 N)(4.00 m) = -80.0 J

Fk

Graphical Representation of Work

• Work can be found by analyzing a plot of force and displacement

• The product F∙d is the area underneath and Fd graph

Graphical Representation of Work

• This is particularly useful when force is not constant (which it normally isn’t)

EnergyEnergy

Section 5.2Section 5.2

Kinetic EnergyKinetic Energy

• Kinetic energy is energy associated with an object in motionmotion

• KE is a scalar quantity• KE depends upon an objects mass and

velocity

• SI unit is the joulejoule

2

2

1mvKE

mN 1 s

m kg 1 J 1

2

2

Sample Problem BSample Problem B

• What is the speed of a 145 g baseball whose kinetic energy is 131 J?

• Show that the solution is dimensionally consistent.

m/s 5.42

kg0.145

J 1312

2

2

1 2

v

v

m

KEv

mvKE

Relationship of Work and EnergyRelationship of Work and Energy

• Work is a transfer of energy• Net work done on an object is equal to the

change in kinetic energy of the object

KEWnet

TheoremEnergy Kinetic-Work

Proof of Proof of W-KE TheoremW-KE Theorem

KEKEKEmvmvW

vvmW

vvxaxmaW

xavvxFW

ififnet

ifnet

ifnet

ifnet

22

22

22

22

2

1

2

1

2

2Δ

2

Importance of Importance of W-KE TheoremW-KE Theorem

• Some problems that can be solved using Newton’s Laws turn out to be very difficult in practice

• Very often they are solved more simply using a different approach…

• An energy approach.

Sample Problem CSample Problem C

• A 10.0 kg sled is pushed across a frozen pond such that its initial velocity is 2.2 m/s. If the coefficient of kinetic friction between the sled and the ice is 0.10, how far does the sled travel? (Only consider the sled as it is already in motion.)

d

vi

Fk

FN

mg

m 5.2

1m/s81.91.02

m/s2.2

)180cos(2

cos21

21

cos

cos

0 m/s; 2.2 0.10; kg; 10.0 :Given

2

2

2

22

d

gμ

vd

mgμ

mvmv

F

KEd

KEdFW

vvμm

k

i

k

if

k

knet

fik

d

Fk

FN

mg

Potential EnergyPotential Energy

• PE is “stored” energy– It has the “potential” to do work

• Energy associated with an object due to its position

• Gravitational PEg

– Due to position relative to earth

• Elastic PEe

– Due to stretch or compression of a spring

Two Types of Potential EnergyTwo Types of Potential Energy

Potential Energy

PEg = mgh

Gravitational Elastic

PEe = ½ kx2

Gravitational Potential EnergyGravitational Potential Energy

• Gravitational PE is energy related to position

PEg = mgh• Gravitational PE is

relative to position• Zero PE is defined by

the problem• If PEc is zero, then

PEA > PEB > PEC

Elastic Potential EnergyElastic Potential Energy

• PE resulting from the compression or stretching of an elastic material or spring.

• PEe = ½ kx2 where…• x = distance

compressed or stretched

• k = spring constant

Spring constant indicates resistance to stretch.

5.3 Conservation of EnergyObjectives

At the end of this section you should be able to

1. Identify situations in which conservation of mechanical energy is valid

2. Recognize the forms that conserved energy can take

3. Solve problems using conservation of mechanical energy

Conserved Quantities

• For conserved quantities, the total remains constant, but the form may change

• Example: one dollar may be changed, but its quantity remains the same.

• Example: a crystal of salt might be ground to a powder, but the mass remains the same. Mass in conserved

Mechanical Energy

• Is conserved in the absence of frictioni.e. initial ME equals final ME

• MEi = MEf

• If ME = KE + PE

• Then KEi + PEi = KEf + PEf

• ½ mvi2 + mghi = ½ mvf

2 + mghf

Conservation of Mechanical Energy( A Falling Egg)

Time (s)

Hght (m)

Spd (m/s)

PE

(J)

KE

(J)

ME

(J)

0.00 1.00 0.00 0.74 0.00 0.74

0.10 0.95 0.98 0.70 0.04 0.74

0.20 0.80 2.00 0.59 0.15 0.74

0.30 0.56 2.90 0.41 0.33 0.74

0.40 0.22 3.90 0.16 0.58 0.74

0.45 0.00 4.43 0.00 0.74 0.74

• As a body falls, potential energy is converted to kinetic energy

• Since ME is conserved (constant)…. ΣPE + KE = ME

•In the absence of friction & air resistance, this is true for mechanical devices also

Conservation of Mechanical Energy

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 0.10 0.20 0.30 0.40 0.50

Time (s)

En

erg

y (

J)

PE

KE

ME

Poly. (PE)

Poly. (KE)

Linear (ME)

Mechanical Energy

• Is the sum of KE and all forms of PE in the system• ME = ΣKE + ΣPE• sigma (Σ ) indicates “the sum of”

Sample Problem 5E• Starting from rest, a child of 25.0 kg slides from a height

of 3.0 m down a frictionless slide. What is her velocity at the bottom of the slide?

• Could solve using kinematic equations, but it is simpler to solve as energy conservation problem.

• MEi = MEf

ME may not be conserved

• In the presence of friction, mechanical energy is not conserved

• Friction converts some ME into heat energy

• Total energy is conserved

MEi = MEf + heat

Work-Kinetic Energy Theorem

• The net work done on an object is equal to the change in kinetic energy of the object

• Wnet = ∆KE

• The work done by friction is equal to the change in mechanical energy

• Wfriction = ∆ME

Power

• Is the rate of work, the rate at which energy is transferred

• P = W/∆t• Since W = Fd, P = Fd /∆t or• P = Fvavg

• Unit of power = the watt (W)1 W = 1 J/s1 hp = 746W horsepower