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Physics for Scientists & Engineers, with Modern

Physics, 4th edition

Giancoli

Piri Reis University 2011-2012/ Physics -I

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Piri Reis University 2011-2012 Fall SemesterPhysics -I

Chapter 7 and 8

Work and Energy

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Piri Reis University 2011-2012Lecture VII and VIII

I. Work Done by a Constant ForceII. Kinetic Energy, and the Work-Energy PrincipleIII. Potential EnergyIV. Conservative and Nonconservative Forces V. The Conservation of Mechanical EnergyVI. Problem Solving Using Conservation of Mechanical EnergyVII. Other Forms of Energy; Energy Transformations and the Law of Conservation of EnergyVIII. Energy Conservation with Dissipative Forces: Solving ProblemsIX. Power

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I. Work Done by a Constant ForceThe work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement:

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W = Fd cosθ

Simple special case: F & d are parallel: θ = 0, cosθ = 1

W = Fd

Example: d = 50 m, F = 30 NW = (30N)(50m) = 1500 N m

Work units: Newton - meter = Joule 1 N m = 1 Joule = 1 J

θ

F

d

I. Work Done by a Constant Force

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I. Work Done by a Constant ForceIn the SI system, the units of work are joules:

As long as this person does not lift or lower the bag of groceries, he is doing no work on it. The force he exerts has no component in the direction of motion.

W = 0:Could have d = 0 W = 0Could have F d θ = 90º, cosθ = 0

W = 0

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Solving work problems:

1. Draw a free-body diagram.

2. Choose a coordinate system.

3. Apply Newton’s laws to determine any unknown forces.

4. Find the work done by a specific force.

5. To find the net work, either

find the net force and then find the work it does, or

find the work done by each force and add.

I. Work Done by a Constant Force

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W = F|| d = Fd cosθm = 50 kg, Fp = 100 N, Ffr = 50 N, θ = 37º

Wnet = WG + WN +WP + Wfr = 1200 J

Wnet = (Fnet)x x = (FP cos θ – Ffr)x = 1200 J

Example

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ExampleΣF =ma

FH = mgWH = FH(d cosθ) = mgh

FG = mgWG = FG(d cos(180-θ)) = -mgh

Wnet = WH + WG = 0

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I. Work Done by a Constant Force

Work done by forces that oppose the direction of motion, such as friction, will be negative.

Centripetal forces do no work, as they are always perpendicular to the direction of motion.

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II. Kinetic Energy, and the Work-Energy Principle

Energy was traditionally defined as the ability to do work. We now know that not all forces are able to do work; however, we are dealing in these chapters with mechanical energy, which does follow this definition.

Energy The ability to do work

Kinetic Energy The energy of motion“Kinetic” Greek word for motion

An object in motion has the ability to do work

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Consider an object moving in straight line. Starts at speed v1. Due to presence of a net force Fnet, accelerates (uniform) to speed v2, over distance d.

II. Kinetic Energy, and the Work-Energy Principle

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Newton’s 2nd Law: Fnet= ma (1)

1d motion, constant a (v2)2 = (v1)2 + 2ad a = [(v2)2 - (v1)2]/(2d) (2)

Work done: Wnet = Fnetd (3)

Combine (1), (2), (3): Wnet = mad = md [(v2)2 - (v1)2]/(2d) OR

Wnet = (½)m(v2)2 – (½)m(v1)2

II. Kinetic Energy, and the Work-Energy Principle

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If we write the acceleration in terms of the velocity and the distance, we find that the work done here is

We define the translational kinetic energy:

II. Kinetic Energy, and the Work-Energy Principle

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This means that the work done is equal to the change in the kinetic energy:

• If the net work is positive, the kinetic energy increases.

• If the net work is negative, the kinetic energy decreases.

II. Kinetic Energy, and the Work-Energy Principle

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Net work on an object = Change in KE.

Wnet = KE The Work-Energy Principle

Note: Wnet = work done by the net (total) force.

Wnet is a scalar.

Wnet can be positive or negative (because KE can be both + & -)

Units are Joules for both work & KE.

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Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules.

Work done on hammer:Wh = KEh = -Fd = 0 – (½)mh(vh)2

Work done on nail:Wn = KEn = Fd = (½)mn(vn)2 - 0

II. Kinetic Energy, and the Work-Energy Principle

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An object can have potential energy by virtue of its surroundings.

Familiar examples of potential energy:

• A wound-up spring

• A stretched elastic band

• An object at some height above the ground

III. Potential Energy

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In raising a mass m to a height h, the work done by the external force is

We therefore define the gravitational potential energy:

WG = -mg(y2 –y1) = -(PE2 – PE1) = -PE

III. Potential Energy

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Wext = PE

The Work done by an external force to move the object of mass m from point 1 to point 2 (without acceleration) is equal to the change in potential energy between positions 1 and 2

WG = -PE

III. Potential Energy

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This potential energy can become kinetic energy if the object is dropped.

Potential energy is a property of a system as a whole, not just of the object (because it depends on external forces).

If , where do we measure y from?

It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.

III. Potential Energy

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Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

III. Potential Energy

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Restoring force

“Spring equation or Hook’s law”

where k is called the spring constant, and needs to be measured for each spring.

(6-8)

The force required to compress or stretch a spring is:

FP = kx

III. Potential Energy

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The force increases as the spring is stretched or compressed further (elastic) (not constant). We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written:

(6-9)

2

2

1)

2

1( kxxkxxFW

III. Potential Energy

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Conservative Force The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass. A PE CAN be defined for conservative forces

Non-Conservative Force The work done by that force depends on the path taken between the initial & final positions of the mass.

A PE CAN’T be defined for non-conservative forces

The most common example of a non-conservative force is FRICTION

IV. Conservative and Nonconservative Forces

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If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

IV. Conservative and Nonconservative Forces

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Potential energy can only be defined for conservative forces.

IV. Conservative and Nonconservative Forces

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Therefore, we distinguish between the work done by conservative forces and the work done by nonconservative forces.

We find that the work done by nonconservative forces is equal to the total change in kinetic and potential energies:

Wnet = Wc + Wnc

Wnet = ΔKE = Wc + Wnc

Wnc = ΔKE - Wc = ΔKE – (– ΔPE)

IV. Conservative and Nonconservative Forces

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If there are no nonconservative forces, the sum of the changes in the kinetic energy and in the potential energy is zero – the kinetic and potential energy changes are equal but opposite in sign.

KE + PE = 0

This allows us to define the total mechanical energy:

And its conservation:

(6-12b)

V. The Conservation of Mechanical Energy

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PE2 = 0 KE2 = (½)mv2

KE1 + PE1 = KE2 + PE2

0 + mgh = (½)mv2 + 0v2 = 2gh

KE + PE = same as at points 1 & 2

The sum remains constant

PE1 = mgh KE1 = 0

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In the image on the left, the total mechanical energy is:

The energy buckets (right) show how the energy moves from all potential to all kinetic.

VI. Problem Solving Using Conservation of Mechanical Energy

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If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting point.

VI. Problem Solving Using Conservation of Mechanical Energy

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For an elastic force, conservation of energy tells us:

VI. Problem Solving Using Conservation of Mechanical Energy

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Some other forms of energy:

Electric energy, nuclear energy, thermal energy, chemical energy.

Work is done when energy is transferred from one object to another.

Accounting for all forms of energy, we find that the total energy neither increases nor decreases. Energy as a whole is conserved.

VII. Other Forms of Energy; Energy Transformations and the Law of Conservation of Energy

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If there is a nonconservative force such as friction, where do the kinetic and potential energies go?

They become heat; the actual temperature rise of the materials involved can be calculated.

VIII. Energy Conservation with Dissipative Forces: Solving Problems

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We had, in general:WNC = KE + PE

WNC = Work done by non-conservative forcesKE = Change in KEPE = Change in PE (conservative forces)

Friction is a non-conservative force! So, if friction is present, we have (WNC Wfr)

Wfr = Work done by frictionIn moving through a distance d, force of friction Ffr does work

Wfr = - Ffrd

VIII. Energy Conservation with Dissipative Forces: Solving Problems

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When friction is present, we have: Wfr = -Ffrd = KE + PE = KE2 – KE1 + PE2 – PE1

– Also now, KE + PE Constant!

– Instead, KE1 + PE1+ Wfr = KE2+ PE2

OR: KE1 + PE1 - Ffrd = KE2+ PE2

• For gravitational PE:

(½)m(v1)2 + mgy1 = (½)m(v2)2 + mgy2 + Ffrd

• For elastic or spring PE: (½)m(v1)2 + (½)k(x1)2 = (½)m(v2)2 + (½)k(x2)2 + Ffrd

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Problem Solving:

1. Draw a picture.

2. Determine the system for which energy will be conserved.

3. Figure out what you are looking for, and decide on the initial and final positions.

4. Choose a logical reference frame.

5. Apply conservation of energy.

6. Solve.

VIII. Energy Conservation with Dissipative Forces: Solving Problems

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Power is the rate at which work is done –

The difference between walking and running up these stairs is power – the change in gravitational potential energy is the same.

(6-17)

In the SI system, the units of power are watts:

British units: Horsepower (hp). 1hp = 746 W

IX. Power

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Power is also needed for acceleration and for moving against the force of gravity.

The average power can be written in terms of the force and the average velocity:

IX. Power

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Summary of Chapter VII and VIII• Work:

•Kinetic energy is energy of motion:

• Potential energy is energy associated with forces that depend on the position or configuration of objects.

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•The net work done on an object equals the change in its kinetic energy.

• If only conservative forces are acting, mechanical energy is conserved.

• Power is the rate at which work is done.

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Example 1 Pulling a Suitcase-on-Wheels

Find the work done if the force is 45.0-N, the angle is 50.0 degrees, and the displacement is 75.0 m.

J 2170

m 0.750.50cosN 0.45cos

sFW

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FssFW 0cos

FssFW 180cos

Example 2

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Example 3 Accelerating a Crate

The truck is accelerating ata rate of +1.50 m/s2. The massof the crate is 120-kg and itdoes not slip. The magnitude ofthe displacement is 65 m.

What is the total work done on the crate by all of the forces acting on it?

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The angle between the displacementand the friction force is 0 degrees.

J102.1m 650cosN180 4W

N180sm5.1kg 120 2 maf s

Example 3 Accelerating a Crate

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Example 4 Deep Space 1

The mass of the space probe is 474-kg and its initial velocityis 275 m/s. If the .056 N force acts on the probe through adisplacement of 2.42×109m, what is its final speed?

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2212

f21W omvmv

sF cosW

Example 4 Deep Space 1

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2

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f219 sm275kg 474kg 474m1042.20cosN 056. v

2212

f21cosF omvmvs

sm805fv

Example 4 Deep Space 1

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In this case the net force is kfmgF 25sin

Example 5

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sFW cos

fo hhmgW gravity

Example 6

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fo hhmgW gravity

Example 7

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Example 8 A Gymnast on a Trampoline

The gymnast leaves the trampoline at an initial height of 1.20 mand reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast?

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2212

f21W omvmv

fo hhmgW gravity

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ofo mvhhmg

foo hhgv 2

sm40.8m 80.4m 20.1sm80.92 2 ov

Example 8 A Gymnast on a Trampoline

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Example 9 A Daredevil Motorcyclist

A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff 38.0 m/s. Ignoring air resistance, findthe speed with which the cycle strikes the ground on the otherside.

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of EE

2212

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ooff mvmghmvmgh

2212

21

ooff vghvgh

Example 9 A Daredevil Motorcyclist

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2212

21

ooff vghvgh

22 ofof vhhgv

sm2.46sm0.38m0.35sm8.92 22 fv

Example 9 A Daredevil Motorcyclist

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Example 10 Fireworks

Assuming that the nonconservative forcegenerated by the burning propellant does425 J of work, what is the final speedof the rocket. Ignore air resistance.

2

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221

oo

ffnc

mvmgh

mvmghW

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2212

21

ofofnc mvmvmghmghW

2

21

2

kg 20.0

m 0.29sm80.9kg 20.0J 425

fv

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fofnc mvhhmgW

sm61fv

Example 10 Fireworks

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Part a) constant speeda = 0 ∑Fx = 0

F – FR – mg sinθ = 0F = FR + mg sinθ

F = 3100 NP = Fv

= 91 hp

Part b) constant accelerationNow, θ = 0∑Fx = ma

F – FR= mav = v0 + at

a = 0.93 m/s2

F = ma + FR = 2000 NP = Fv

= 82 hp

Example 11

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HOMEWORK

Giancoli, Chapter 7

3, 7, 8, 11, 14, 25, 32, 40, 45, 47Giancoli, Chapter 8

7, 9, 10, 17, 20, 21, 23, 28, 36, 38 Referenceso “Physics For Scientists &Engineers with Modern Physics” Giancoli 4th edition, Pearson International Edition