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Status: Unit 3Status: Unit 3

Force, Mass, & NewtonForce, Mass, & Newton’’s 1s 1stst, 2, 2ndnd, 3, 3rdrd Laws Laws of Motion (4of Motion (4--1 through 41 through 4--5)5)Weight Weight –– the Force of Gravity: and the the Force of Gravity: and the Normal Force, Problem solving (4Normal Force, Problem solving (4--6, 46, 4--8, 8, 44--7)7)

•• Solving Problems w/ NewtonSolving Problems w/ Newton’’s Laws, s Laws, Problems w/ Friction (4Problems w/ Friction (4--7, 57, 5--1)1)

•• Problems w/ Friction and Terminal Velocity Problems w/ Friction and Terminal Velocity revisited (5revisited (5--1, 51, 5--5)5)

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Free Body DiagramsFree Body Diagrams

•• A key element in understanding motionA key element in understanding motion•• A freeA free--body diagram has:body diagram has:

1.1. A convenient coordinate system,A convenient coordinate system,2.2. Representative vectors Representative vectors

I.I. For For allall forces acting forces acting onon a bodya bodyII.II. Including those that are unknown.Including those that are unknown.III.III. If translational only, at the center of the body.If translational only, at the center of the body.

3.3. Descriptive labels for each force vector.Descriptive labels for each force vector.

•• DoesnDoesn’’t show forces the body exerts on other t show forces the body exerts on other objects.objects.

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The PlanThe Plan

•• The best way to get a handle on the analysis of The best way to get a handle on the analysis of motion using Newtonmotion using Newton’’s laws is by doing s laws is by doing problems in class and at home.problems in class and at home.

•• With each problem we add complexity and new With each problem we add complexity and new forces.forces.

•• So far weSo far we’’ve encounteredve encountered–– one fundamental force (gravity)one fundamental force (gravity)–– one nonone non--fundamental force (the normal force).fundamental force (the normal force).

•• Today weToday we’’ll introduce two more nonll introduce two more non--fundamental forcesfundamental forces–– TensionTension–– Friction.Friction.

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Back to Mechanics: A Rope under TensionBack to Mechanics: A Rope under Tension

•• A flexible rope, cord, or wire pulling on an object is said A flexible rope, cord, or wire pulling on an object is said to be under tension and exerts a force to be under tension and exerts a force FFTT. .

•• Before we begin, we make a simple assumption that any Before we begin, we make a simple assumption that any such device is massless.such device is massless.

•• As a consequence the rope transmits force undiminished As a consequence the rope transmits force undiminished from one end to the other. from one end to the other.

•• This is apparent from This is apparent from ΣΣFF = m= maa. Since the mass is zero . Since the mass is zero the net force on the cord is zero, so the force on the two the net force on the cord is zero, so the force on the two ends must sum to zero. ends must sum to zero.

•• Where does this other force come, from the object Where does this other force come, from the object pulled!pulled!

FT-FT

FT

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An Example: Two Boxes and a Cord.An Example: Two Boxes and a Cord.

•• Two boxes resting on a Two boxes resting on a frictionless surface are frictionless surface are connected by a massless connected by a massless cord.cord.

•• The boxes have masses The boxes have masses of 12.0 and 10.0 kg. of 12.0 and 10.0 kg.

•• A horizontal force of A horizontal force of FP=40.0 N is applied by FP=40.0 N is applied by pulling on the lighter pulling on the lighter box.box.

•• What is the acceleration What is the acceleration of each box and the of each box and the tension in the cord? tension in the cord?

•• This problem adds a This problem adds a force and 2 dimensions!force and 2 dimensions!

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•• Box 1 freebody diagram Box 1 freebody diagram has four forces:has four forces:–– From person pulling: From person pulling: FFPP

–– Tension from cord: Tension from cord: FFTT

–– Weight: Weight: WW11 = m= m11gg–– Normal Force: Normal Force: FFN1N1

•• From the 2From the 2ndnd Law in the Law in the x directionx direction

11amFFF TPx =−=Σ

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•• Box 2 freebody diagram Box 2 freebody diagram has three forces:has three forces:–– Tension from cord: Tension from cord: FFTT

–– Weight: Weight: WW22 = m= m22gg–– Normal Force: Normal Force: FFN2N2

•• From the 2From the 2ndnd Law in the Law in the x directionx direction

22amFF Tx ==Σ

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•• The boxes must have identical acceleration The boxes must have identical acceleration aa11==aa22 otherwise the otherwise the cord would part or bunchcord would part or bunch--up which is counter to our experience.up which is counter to our experience.

•• It makes sense for the tension to be less than the pulling forceIt makes sense for the tension to be less than the pulling forcesince the tension accelerates only the second boxsince the tension accelerates only the second box

•• Note how analyzing two freeNote how analyzing two free--body diagrams allows us to body diagrams allows us to understand an understand an ““innerinner”” force.force.

NsmkgamF

smkgN

mmFa

ammFamamFFF

amamFamamFF

T

P

P

TTP

TTP

8.21)/82.1)(0.12(

/82.10.220.40

)(

:equations two theAdding

2Box 1Box

22

2

21

21

21

222111

===

==+

=

→+=+=+−

====−

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AtwoodAtwood’’s Machines Machine

•• A classic problem: Two A classic problem: Two suspended masses suspended masses connected by a cable connected by a cable passing over a pulley. passing over a pulley.

•• Consider an elevator of Consider an elevator of mass mmass m11 and a counterand a counter--weight of mass mweight of mass m22, where , where mm1 1 > m> m22..

•• In terms of the masses In terms of the masses (to keep it general) (to keep it general) calculate the acceleration calculate the acceleration of the elevator and the of the elevator and the tension in the cable. tension in the cable.

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The Rev. George The Rev. George Atwood (1746Atwood (1746--1807) 1807) was a tutor at Trinity was a tutor at Trinity College, Cambridge College, Cambridge when he published when he published ““A A Treatise on the Treatise on the Rectilinear Motion and Rectilinear Motion and Rotation of Bodies, with Rotation of Bodies, with a Description of Original a Description of Original Experiments Relative to Experiments Relative to the Subjectthe Subject”” in 1784. in 1784.

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•• Once again in order to Once again in order to avoid stretching or avoid stretching or bunching of the cable the bunching of the cable the two masses must have two masses must have equal & opposite equal & opposite acceleration: acceleration: aa11==--aa22

•• A massless cable also A massless cable also ensures that the tension ensures that the tension is equal on both sides of is equal on both sides of the cable or on both the cable or on both objects.objects.

•• Since the elevator mass Since the elevator mass is greater than the CW is greater than the CW the elevator will be the elevator will be moving down.moving down.

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•• Limiting cases are Limiting cases are correct:correct:–– Masses equal: Masses equal: aa=0 or =0 or

it doesnit doesn’’t move!t move!–– Second mass zero: Second mass zero:

a=g and the elevator a=g and the elevator is in free fall.is in free fall.

gmmmma

ammgmmamamgmgm

a)(-mamg)m(FgmF

amamgm Fa -mamgmF

TT

TT

)()(

)()(

:eq.CW thefrom eq. Elev heSubtract t

:ghtCounterwei :Elevator

21

21

2121

2121

1212

22221111

+−

=

→+=−→+=−

→−=−−−

==−==−

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•• We can use the equation for We can use the equation for acceleration to derive the tensionacceleration to derive the tension

•• Some limiting casesSome limiting cases–– Masses equal: FMasses equal: FTT = mg =W= mg =W–– Either mass zero: tension is zero Either mass zero: tension is zero

and weand we’’re in free fall again!re in free fall again!

•• Note how the algebraic form Note how the algebraic form helps us understand the physics.helps us understand the physics.

→+−

−=

→+−

−=

−=→=→=−

+−

=

==−

gmmmmmF

gmmmmgmF

agmFa-mgmF

a-mgmF

gmmmma

a -mamgmF

T

T

T

T

T

T

))()(1(

) )()((

afor ngSubstituti)(

equationfirst thegRearrangin

)()( :inAccelerato

:Elevator

21

211

21

211

1

111

11

21

21

1111

gmmmmF

gmm

mmmmmF

gmmmm

mmmmmF

T

T

T

+

=

→

+

−−+=

→

+−

−++

=

)(2

)()()(

)()(

)()(

21

21

21

21211

21

21

21

211

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The mechanical advantage of a pulleyThe mechanical advantage of a pulley

•• Apply the 2Apply the 2ndnd Law to Law to the freebody diagram the freebody diagram at zero acceleration.at zero acceleration.

•• In the y directionIn the y direction

•• So for N loops the So for N loops the force required is force required is mg/N!mg/N!

2/02

2

mgFmgF

mamgF

T

T

T

=→=−→=−

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The Mechanical Advantage of Two The Mechanical Advantage of Two Dimensional TensionDimensional Tension

•• Because tension is always a long a rope (otherwise it Because tension is always a long a rope (otherwise it would buckle) pushing in the middle applies a much would buckle) pushing in the middle applies a much stronger force on the ends.stronger force on the ends.

•• We can show this with We can show this with ΣΣFF=m=maa..•• In the figure below if FIn the figure below if FPP=300 N, what force is pulling on =300 N, what force is pulling on

the vehicle (and tree)?the vehicle (and tree)?

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•• LetLet’’s look at the x direction:s look at the x direction:

•• In other words the magnitudes of the forces on the In other words the magnitudes of the forces on the rope at the fulcrum are equal.rope at the fulcrum are equal.

•• Looking at the y directionLooking at the y direction

21

21

21

0)cos()cos(so 0,a budgescar thebeforeJust

)cos()cos(

TT

TT

TTx

FFFF

FFF

=→=−=

−=Σ

θθ

θθ

NNFF

FF

FFF

oP

T

TP

TPy

1700)5sin(2

300)sin(2

0)sin(2

)sin(2

=≈=

→=−

→−=Σ

θ

θ

θThe technique increasesthe force on the car six-fold. Note the smallerθ the larger the effect!

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A List of ForcesA List of Forces

•• Fundamental ForcesFundamental Forces–– Gravitational ForceGravitational Force –– the the

oldest, weakest, and least oldest, weakest, and least understoodunderstood

–– Electroweak Force Electroweak Force –– a a unification ofunification of

•• Electromagnetism (electric Electromagnetism (electric and magnetic)and magnetic)

•• Weak force (radioactive Weak force (radioactive force)force)

–– The Strong Force The Strong Force –– the the nuclear force and strongest nuclear force and strongest of the three.of the three.

•• Non Fundamental ForcesNon Fundamental Forces–– Normal ForceNormal Force–– FrictionFriction

•• StaticStatic•• KineticKinetic

–– TensionTension

•• Most nonMost non--fundamental fundamental forces are realforces are real--world world manifestations of the manifestations of the gravitational and gravitational and electroweak force.electroweak force.

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Origin of Forces: Standard Model of Origin of Forces: Standard Model of Particle PhysicsParticle Physics•• The essence: Bits of matter stick together by exchanging The essence: Bits of matter stick together by exchanging

stuff.stuff.•• The great achievement of particle physics is a model that The great achievement of particle physics is a model that

describes all particles and particle interactions. The model describes all particles and particle interactions. The model includes:includes:–– 6 quarks (the particles in the nucleus) and their 6 quarks (the particles in the nucleus) and their

antiparticles.antiparticles.–– 6 leptons (of which the electron is an example) and 6 leptons (of which the electron is an example) and

their antiparticlestheir antiparticles–– 4 force carrier particles 4 force carrier particles

•• Precisely: Precisely: ““All known matter is All known matter is composed of composites of quarks composed of composites of quarks and leptons which interact by and leptons which interact by exchanging force carriers.exchanging force carriers.””

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Explained by Standard Model10-37 weakerthan EM, not

explained

The Stuff: Standard Model Interactions Mediated The Stuff: Standard Model Interactions Mediated by Boson Exchangeby Boson Exchange

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The Bits: Periodic Table of Fundamental ParticlesThe Bits: Periodic Table of Fundamental Particles

Mass

All point-like down to10-18 m

Families reflectincreasing mass and

a theoreticalorganization

u, d, n, e are “normal matter”

These all interact by exchanging bosons

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Two more problems, First the Two more problems, First the AccelerometerAccelerometer

•• We can apply the 2We can apply the 2ndnd law to build a crude accelerometer. law to build a crude accelerometer. Consider the scenario below.Consider the scenario below.

•• When the car is accelerating the pendulum makes an When the car is accelerating the pendulum makes an angle,angle, θθ, with the vertical., with the vertical.

•• How does How does θθ depend on the acceleration?depend on the acceleration?

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•• There are two forces, There are two forces, tension in the string, tension in the string, FFT,T,and the weight, mand the weight, mg.g.

•• Since there is no Since there is no accelaccel--erationeration in the vertical in the vertical direction:direction:

•• And in And in thexthex direction there direction there is an acceleration, thusis an acceleration, thus

•• An acceleration of 1.2m/sAn acceleration of 1.2m/s22

= (2.6mi/hr)/s would = (2.6mi/hr)/s would correspond to 7.0correspond to 7.0oo

)tan(

)tan(

)cos()sin(

:first by theequationlast thisDividing

)sin(

θ

θ

θθ

θ

gaga

mgma

FF

FmaF

T

T

Tx

=

→=

→=

==Σ

mgF

mgFF

T

Ty

=

→−==Σ

)cos()cos(0

θ

θ

θ

θ

FTcos(θ)

+y

+x

Note this accelerometeris independent of tensionand mass.

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A box on an incline A box on an incline •• This is another classic problem which we will make more This is another classic problem which we will make more

realistic next lecture with friction. For now assume an realistic next lecture with friction. For now assume an ideal surface.ideal surface.

•• A box of mass m is on an inclined plane that makes an A box of mass m is on an inclined plane that makes an angle angle θθ with the horizontal. with the horizontal. –– What is the normal force? What is the acceleration?What is the normal force? What is the acceleration?

•• Evaluate for a mass m=10kg and an angle of Evaluate for a mass m=10kg and an angle of θθ=30=30oo..

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•• Since the box never Since the box never ““lifts lifts offoff”” the incline we put the incline we put the x axis for our freethe x axis for our free--body problem along the body problem along the incline. This simplifies incline. This simplifies the problem since there the problem since there will be no acceleration in will be no acceleration in the y direction.the y direction.

•• We then have two We then have two forces:forces:–– The normal force to the The normal force to the

planeplane–– Gravity which as shown in Gravity which as shown in

the figure we resolve into the figure we resolve into x and y x and y compomentscompoments..

•• Next we apply Next we apply ΣΣF=maF=ma

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•• First in the y direction:First in the y direction:

In the limit:In the limit:–– No angle the normal force No angle the normal force

is the weight as expected.is the weight as expected.–– At 90 degrees, FAt 90 degrees, FNN=0, the =0, the

incline is straight up and incline is straight up and we are in freewe are in free--fall, no fall, no normal forcenormal force..

•• For the values givenFor the values given

•• Now the x direction:Now the x direction:

•• The acceleration is The acceleration is always less than g.always less than g.

•• In the limitIn the limit–– No angle the acceleration No angle the acceleration

is zero.is zero.–– At 90 degrees, a=g, the At 90 degrees, a=g, the

incline is straight up, and incline is straight up, and we are in freewe are in free--fallfall

•• For the values givenFor the values given

)sin()sin(

θθ

gamgmaF

x

xx

=→==Σ

)cos()cos(0

θ

θ

mgF

mgFF

N

Ny

=

→−==Σ

Nsmkg

mgFN

85)30)(cos/8.9)(10(

)cos(2 =

== θ

2

2

/9.4

)30sin()/8.9(

sm

smax

=

=

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•• WeWe’’ve solved a number of interesting ve solved a number of interesting problems:problems:–– Boxes under tensionsBoxes under tensions–– AtwoodAtwood’’s machines machine–– Cables as levelsCables as levels–– AccelerometerAccelerometer–– Box on an inclineBox on an incline

•• Although simple these problems show the Although simple these problems show the great power of Newtongreat power of Newton’’s Lawss Laws

•• Next lecture weNext lecture we’’ll expand the scope to ll expand the scope to encompass friction.encompass friction.