Work…Work… In everyday speech work In everyday speech work

has a very general has a very general meaning. In describing meaning. In describing motion in physics, work motion in physics, work has a very specific has a very specific meaning.meaning.

Chair ExampleChair Example

StandingStanding WalkingWalking

No work is done on the No work is done on the chairchair

Work is defined as the Work is defined as the productproduct of the of the force appliedforce applied to cause to cause motion and the motion and the distance the distance the object movesobject moves in the in the direction direction of the forceof the force..

Work is done only when Work is done only when components of a force are components of a force are parallel to a displacementparallel to a displacement

FORMULAFORMULA

W = fdW = fdIN DIRECTION OF MOTIONIN DIRECTION OF MOTION

The symbol for work is WThe symbol for work is W Work has 2 acceptable unitsWork has 2 acceptable units

Nm Nm Joules (J)Joules (J)

Lifting an apple Lifting an apple about 2ft is a Jouleabout 2ft is a Joule

3 good push-ups is 3 good push-ups is about 1000Jabout 1000J

JOULEJOULE

WORKIn order for work to be done, three things are necessary:•There must be an applied force.•The force must act through a certain distance, called the displacement.•The force must have a component along the displacement.

Work is a scalar quantity equal to the product of the magnitudes of the displacement and the component of the force in the direction of the displacement.W = F . x or W = F cos x

 UNITS: N.m this unit is called a Joule (J)

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.

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.

If the force acting on an object varies in magnitude and/or direction during the object’s displacement, graphical analysis can be used to determine the work done. The Force is plotted on the y-axis and the distance through which the object moves is plotted on the x-axis. The work done is represented by the area under the curve.

5.1 A push of 200 N moves a 100 N block up a 30inclined plane. The coefficient of kinetic friction is 0.25 and the length of the plane is 12 m.a. Find the work done by each force acting on the block.

FA = 200 NFG = 100 Nθ = 30˚μ = 0.25x = 12 m

Forces acting: Ff FA FG and FN

FN does NO work.

W

FN

Ff

FG

FGy

FGx

θ

W FA = FA x = 200 (12) = 2400 J

Ff = μ FN

= μ FG cos 30˚ = 0.25 (100) cos 30˚ = 21.6 N

WFf = - Ff x = - 21.6 (12) = - 259.2 J

WFG = FG x = -FGx x = - FG

sin30˚x

= - 100 sin 30˚ (12) = - 600 J

FA

FN

Ff

FG

FGy

FGx

θ

b. Show that the net work done by these forces is the same as the work of the resultant force.

Net work: ΣW = 2400 - 259.2 - 600 = 1540.8 JThe resultant force: ΣFx = FA - Ff - FGx

= 200 - 21.6 - 50 = 128.4 NWF = Fx . x = 128.4 (12) = 1540.8 J

FA

Positive WorkPositive Work

Negative WorkNegative Work

No WorkNo Work

No WorkNo Work

Force is in the Force is in the direction of direction of

motionmotionForce opposes Force opposes

motionmotion

Force is 90Force is 90°° to to motionmotion

Object is not in Object is not in motionmotion

Situations that Situations that affect the sign of affect the sign of

workwork

1010NN

Moves 2mMoves 2m

W = fdW = fd

W = 20J of WorkW = 20J of Work

Notice direction Notice direction of motion is the of motion is the

same as the same as the applied forceapplied force

6060°°

10N10N

XX

YY

2m2m

How would you How would you solve this? Force solve this? Force applied is NOT in applied is NOT in

the same the same direction as the direction as the objects motion.objects motion.

Think back to vectors and use the Think back to vectors and use the component of the force applied in component of the force applied in

the direction the object moves.the direction the object moves.

6060°°

10N10N

XX

YY

2m2m

COS COS θ = adj/ hypθ = adj/ hyp

COS 60° = force parallel to COS 60° = force parallel to motion motion

10N10N

force para. =force para. =

COS 60COS 60°° (10N) (10N)

force para. = 5Nforce para. = 5N

w = F(parallel) Dw = F(parallel) D

5N (2m) = 10Nm5N (2m) = 10Nm

W = Fd (COS W = Fd (COS θ)θ)Always measure angle with Always measure angle with

horizontal!horizontal!

The above formula works in every The above formula works in every casecaseθ = θ =

0°0°

θ = 90°θ = 90°No work because No work because

no motion in no motion in direction of forcedirection of force

WorkThe VERTICAL component of the force DOES NOT cause the block to move the right. The energy imparted to the box is evident by its motion to the right. Therefore ONLY the HORIZONTAL COMPONENT of the force actually creates energy or WORK.

When the FORCE and DISPLACEMENT are in the SAME DIRECTION you get a POSITIVE WORK VALUE. The ANGLE between the force and displacement is ZERO degrees. What happens when you put this in for the COSINE?

When the FORCE and DISPLACEMENT are in the OPPOSITE direction, yet still on the same axis, you get a NEGATIVE WORK VALUE. This negative doesn't mean the direction!!!! IT simply means that the force and displacement oppose each other. The ANGLE between the force and displacement in this case is 180 degrees. What happens when you put this in for the COSINE?

When the FORCE and DISPLACEMENT are PERPENDICULAR, you get NO WORK!!! The ANGLE between the force and displacement in this case is 90 degrees. What happens when you put this in for the COSINE?

The Work Energy TheoremUp to this point we have learned Kinematics and

Newton's Laws. Let 's see what happens when we apply BOTH to our new formula for WORK!

1. We will start by applying Newton's second law!

2. Using Kinematic #3!3. An interesting term

appears called KINETIC ENERGY or the ENERGY OF MOTION!

The Work Energy Theorem

And so what we really have is called the WORK-ENERGY THEOREM. It basically means that if we impart work to an object it will undergo a CHANGE in speed and thus a change in KINETIC ENERGY. Since both WORK and KINETIC ENERGY are expressed in JOULES, they are EQUIVALENT TERMS!

" The net WORK done on an object is equal to the change in kinetic energy of the object."

Example W=FxcosA 70 kg base-runner begins to slide into second base when

moving at a speed of 4.0 m/s. The coefficient of kinetic friction between his clothes and the earth is 0.70. He slides so that his speed is zero just as he reaches the base (a) How much energy is lost due to friction acting on the runner? (b) How far does he slide?

)8.9)(70)(70.0(

mgFF nf

= 480.2 N

f

of

f

W

mvW

KWa

22 )4)(70(21

210

)

-560 J

x

x

xFW ff

)180(cos)2.480(560

cos

1.17 m

EnergyEnergy

The Stuff that makes The Stuff that makes things movethings move

The ability to do workThe ability to do work Has the units of Joules (J)Has the units of Joules (J) There are 2 kinds of There are 2 kinds of mechanical energymechanical energy

Kinetic EnergyKinetic Energy This is the energy associated with an This is the energy associated with an

objects motion.objects motion. KE depends on mass and velocityKE depends on mass and velocity When the object is treated as a particle, When the object is treated as a particle,

the formula for KE is…the formula for KE is…

KE = ½ mVKE = ½ mV22

manipulatedmanipulated

V = 2KE/m M = V = 2KE/m M = 2KE/V2KE/V22

KE is a scalar quantityKE is a scalar quantity The SI unit for KE is the Joule, The SI unit for KE is the Joule,

yes the same as for workyes the same as for work Look at sample prob. 5BLook at sample prob. 5B

Page 173Page 173 DO practice problemsDO practice problems 5B 1-5 on page 1745B 1-5 on page 174

KE is a scalar quantityKE is a scalar quantity The SI unit for KE is the Joule, The SI unit for KE is the Joule,

yes the same as for workyes the same as for work Look at sample prob. 5BLook at sample prob. 5B

Page 173Page 173 DO practice problemsDO practice problems 5B 1-5 on page 1745B 1-5 on page 174

Work- Kinetic Energy Work- Kinetic Energy TheoremTheorem

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

WWnetnet = = ΔΔKEKE WWnetnet = KE = KEfinalfinal – KE – KEinitial initial

fd(cos fd(cos θθ) = ½ mV) = ½ mV22

The KE of an object is equal to the The KE of an object is equal to the work that moving object can dowork that moving object can do

This theorem allows us to think This theorem allows us to think of KE as the work an object can of KE as the work an object can do as it comes to rest, or the do as it comes to rest, or the amount of energy contained in amount of energy contained in the moving objectthe moving objectThe KE of the moving The KE of the moving

hammer can do workhammer can do work

KE = Work done KE = Work done (net)(net)

fd = ½ mvfd = ½ mv22some of the energy is sound, some of the energy is sound,

heat and light (if spark)heat and light (if spark)

Potential EnergyPotential Energy This is the energy associated This is the energy associated

with an object due to the position with an object due to the position of the object.of the object.

STORED ENERGYSTORED ENERGY There are two kinds of potential There are two kinds of potential

energyenergy1.1. GRAVITATIONAL POTENTIAL GRAVITATIONAL POTENTIAL

ENERGYENERGY2.2. ELASTIC POTENTIAL ENERGYELASTIC POTENTIAL ENERGY

Gravitational Potential Gravitational Potential Energy (PEEnergy (PEgg))

The energy associated with an The energy associated with an object due to the objects object due to the objects position relative to a position relative to a gravitational referencegravitational referenceWh = PEWh = PEgg = mgh = mgh

= mass x gravity x height = mass x gravity x height accelerationacceleration

Has the unit of Has the unit of joulesjoules

gm = wgm = w

Elastic Potential EnergyElastic Potential Energy(PE(PEelasticelastic))

The energy associated The energy associated with a stretched or with a stretched or compressed elastic compressed elastic objectobject Spring, bungee cord, Spring, bungee cord, rubber bandrubber band

Elastic Potential Energy

Overhead Overhead (springs)(springs)

In both the compressed and In both the compressed and stretched example, energy is stretched example, energy is storedstored

PEPEelasticelastic = ½ KX = ½ KX22

K = spring constantK = spring constant X = distance stretched or X = distance stretched or

compressedcompressedPractice Practice

Problems 5D 1-Problems 5D 1-3 3

pg. 180pg. 180

Energy is transferred from one Energy is transferred from one form to anotherform to another

PendulumPendulum

PE = maxPE = max

KE = minKE = minPE = minPE = min

KE = maxKE = max

PE = maxPE = max

KE = minKE = min

As the pendulum As the pendulum swings, PE is swings, PE is

transferred to KE. As transferred to KE. As the bob swings the bob swings

upwards KE is stored as upwards KE is stored as PEPE

10M10M

PE = 10 JPE = 10 J

KE = 0 JKE = 0 J

PE = 0 JPE = 0 J

KE = 10 JKE = 10 J

PE = 5 JPE = 5 J

KE = 5 JKE = 5 J

PE = PE = mghmgh

A falling eggA falling eggMass = .1kg Height = 10mMass = .1kg Height = 10m

ENERGY

Energy is that which can be converted into work. When something has energy, it is able to perform work or, in a general sense, to change some aspect of the physical world.

In mechanics we are concerned with two kinds of energy: KINETIC ENERGY: K, energy possessed by a body by virtue of its motion.

K mv1

2

2 Units: Joules (J)

POTENTIAL ENERGY: PE, energy possessed by a system by virtue of position or condition.

PE = m g h Units: Joules (J)

WORK-ENERGY PRINCIPLE: The work of a resultant external force on a body is equal to the change in kinetic energy of the body.

W = K Units: Joules (J)

W = PE

5.2 What average force F is necessary to stop a 16 g bullet traveling at 260 m/s as it penetrates into wood at a distance of 12 cm?

vf = 0 m/sm = 0.016 kgvo = 260 m/sx = 0.12 m

W = ΔK )(2

1 22of vvm

)12.0(2

)260)(016.0( 2

= - 4506.7 N

WE

21

2 oFx mv

2

2omv

Fx

CONSERVATIVE AND NON-CONSERVATIVE FORCES

The work done by a conservative force depends only on the initial and final position of the object acted upon. An example of a conservative force is gravity.

The work done equals the change in potential energy and depends only on the initial and final positions above the ground and NOT on the path taken.

Friction is a non-conservative force and the work done in moving an object against a non-conservative force depends on the path. For example, the work done in sliding a box of books against friction from one end of a room to the other depends on the path taken.

For mechanical systems involving conservative forces, the total mechanical energy equals the sum of the kinetic and potential energies of the objects that make up the system and is always conserved.

PE K

A roller-coaster car moving without friction illustrates the conservation of mechanical energy.

In real life applications, some of the mechanical energy is lost due to friction. The work due to non-conservative forces is given by:  WNC = ΔPE + ΔK

orWNC = Ef - Eo

5.3 A ballistic pendulum apparatus has a 40-g ball that is caught by a 500-g suspended mass. After impact, the two masses rise a vertical distance of 45 mm. Find the velocity of the combined masses just after impact.

m1 = 0.04 kgm2 = 0.500 kgh = 0.045 m

K0 = PEf

COE

PEf = (m1+m2) ghf

= (0.04+0.500)(9.8)(0.045) = 0.24 J K0 = PEf = 0.24 J

K0 = PEf m1 = 0.04 kgm2 = 0.500 kgh = 0.045 m

20 0

1

2 TK m v

2(0.24)

0.540

2 o

T

Kv

m = 0.94 m/s

5.4 The tallest and fastest roller coaster in the world is the Steel Dragon in Japan. The ride includes a vertical drop of 93.5 m. The coaster has a speed of 3 m/s at the top of the drop. a. Neglect friction and find the speed of the riders at the bottom.? vA = 3 m/s

hA = 93.5 mhB = 0 m

At point A: PEA + KA

At point B: KB

PEA + KA= KB22

2

1

2

1BA mvmvmgh

ghvv AB 22

A

B

2(3) 2(9.8)(93.5) = 42.9 m/s (about 96 mi/h)

COE

b. Find the work done by non-conservative forces on a 55 kg rider during the descent if the actual velocity at the bottom is 41 m/s.

WNC = Ef - E0

= KB - (PEA + KA)

vA = 3 m/svB = 41 m/shA = 93.5 mhB = 0 mm = 55 kg2 21 1

2 2B A Amv mgh mv

2 21 1(55)(41) (55)9.8(93.5) (55)(3)

2 2

= - 4416.5 J

5.5 A 20-kg sled rests at the top of a 30˚ slope 80 m in length. If μk= 0.2, what is the velocity at the bottom of the incline?m = 20

kgθ = 30°r = 80 mμk= 0.2

WNC = Ef - Eo

= Kf - PE0

COE

m = 20 kgθ = 30°x = 80 mμk= 0.2

xh

h = x sin θh = 80 sin 30° = 40 m PE0 = mgh0

= 20(9.8)(40) = 7840 J

Ff = μkFN = μk Fgy = μk Fgcos30° = (0.2)(20)(9.8)cos30° = 34 N

WNC = - Ff r = - 34 (80) = - 2720 J

WNC= Kf - PE0 Kf = PE0 + WNC

= 7840 - 2720 = 5120 J

21

2f fK mv

2f

Kv

m 2(5120)

20 = 22.6

m/s

Mechanical EnergyMechanical Energy The sum of Kinetic Energy The sum of Kinetic Energy

and ALL forms of Potential and ALL forms of Potential energy associated with an energy associated with an object or group of objectsobject or group of objects

ME is not a unique form of ME is not a unique form of energy. Its merely a way of energy. Its merely a way of classifying energyclassifying energy

ME includes KE and PEME includes KE and PE

Mechanical EnergyMechanical Energy ME is different from non ME is different from non

mechanical energy (nuclear, mechanical energy (nuclear, chemical, thermal, internal, chemical, thermal, internal, electrical)electrical)

ME = ME = ΣΣ KE + KE + Σ PEΣ PE ME = ½ mvME = ½ mv22 + mgh + mgh (if PE is NOT present, (if PE is NOT present, elastic)elastic)

Σ SPN

Conservation Of Conservation Of Mechanical EnergyMechanical Energy

Conservation of Mechanical Conservation of Mechanical Energy can also be written Energy can also be written as…as… MEMEii = ME = MEff

½ mv½ mvii22 + mgh + mghi i = ½ mv= ½ mvff

22 + mgh + mghff

True when friction can be True when friction can be ignoredignored

The Law of Conservation of Energy:The Law of Conservation of Energy: The total energy of a closed system The total energy of a closed system is constant.is constant.

Often is the case that KEOften is the case that KEii or or KEKEff or PE or PEii or PE or PEff will be zero. will be zero. When that is the case…When that is the case…

mgh = ½ mvmgh = ½ mv22

2mgh = v2mgh = v22

mm

V = V = 2gh2gh

h = Vh = V22

2g2g

PowerPower This quantity also has a very This quantity also has a very

specific meaning in science that specific meaning in science that can be confused by common can be confused by common English usageEnglish usage

Power is the rate of doing workPower is the rate of doing work That is to say that power is the That is to say that power is the rate at which energy is rate at which energy is transferredtransferred

PowerPower Power is work done divided by the Power is work done divided by the

time taken to do the worktime taken to do the work

Power = WorkPower = Work = fd= fd P = wP = w

Time Time tt tt Power is measured in watts (W) J/sPower is measured in watts (W) J/s A watt is a small unit, 1 watt is about A watt is a small unit, 1 watt is about

what is needed to lift a 2N glass of what is needed to lift a 2N glass of water .5m to your mouth in 1 second.water .5m to your mouth in 1 second.

WattsWatts Since watts are so small, we Since watts are so small, we sometimes use Kilowattssometimes use Kilowatts 1 KW = 1000W1 KW = 1000W

Watts are metric Watts are metric Horse power is traditionalHorse power is traditional 1 Horse power = 746 Watts1 Horse power = 746 Watts

WattsWatts

Watts are named after Watts are named after James Watt, the inventor James Watt, the inventor of the steam engineof the steam engine

Practice ProblemPractice Problem An electric motor lifts an elevator An electric motor lifts an elevator

that weighs 12000N a distance of that weighs 12000N a distance of 9m in 15sec9m in 15sec What is the motors power in watts?What is the motors power in watts? What is the motors power in What is the motors power in

kilowatts?kilowatts?GivenGiven

f = 12000Nf = 12000N

d = 9md = 9m

t = 15st = 15s

P = ?P = ?

FormulaFormula

P = fd/tP = fd/tSolution

P = 12000(9)

15A.A. P = 7200 WP = 7200 W

B.B. 7.2 KW7.2 KW

POWERIs the rate at which work is performed.P = work/time

The difference between walking and running up these stairs is power.

The change in gravitational potential energy is the same.

W FrP Fv

t t

UNITS: J

=Ws

= Watt

5.6 A 1100-kg car starting from rest, accelerates for 5.0 s. The magnitude of the acceleration is 4.6 m/s2. What power must the motor produce to cause this acceleration? m = 1100

kgvo = 0 m/s

t = 5 s

a = 4.6 m/s2

F = ma = (1100)(4.6) = 5060 N

vf = vo + at = 0 + 4.6 (5) = 23 m/s The average velocity is: 23/2 = 11.5 m/s 

P = Fv = (5060)(11.5) = 5.82x104 W

ELASTIC FORCEThe force Fs applied to a spring to stretch it or to compress it an amount x is directly proportional to x.

Fs = - k x Units: Newtons (N)

Where k is a constant called the spring constant and is a measure of the stiffness of the particular spring. The spring itself exerts a force in the opposite direction:

This force is sometimes called restoring force because the spring exerts its force in the direction opposite to the displacement. This equation is known as the spring equation or Hooke’s Law.

1

2A bh

1( )( )

2x kx

21

2kx

The elastic potential energy is given by: PEs = ½ kx2 Units: Joules (J)

5.7 A dart of mass 0.100 kg is pressed against the spring of a toy dart gun. The spring (k = 250 N/m) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its normal length, what speed does the dart acquire?m = 0.1 kg

k = 250 N/mx = 0.06 mPEs = K½ kx2 = ½ mv2

m

kxv

2

1.0

)06.0(250 2

= 3 m/s

page 108 Problem 4