Newton’s Laws - continued

Friction, Inclined Planes, N3L, Law of Gravitation

TWO types of Friction

Static – Friction that keeps an object at rest and prevents it from moving

Kinetic – Friction that acts during motion

Force of Friction The Force of Friction is

directly related to the normal force.

Mostly due to the fact that BOTH are surface forces

Nkkf

Nssf

Nf

FμF

FμF

FF

Note: Friction depends ONLY on the MATERIALS sliding against each other, NOT on surface area!

•μ - coefficient of friction - a unitless constant that is specific to the material type•μ usually has a value of less than one.

ExampleA 1500 N crate is being pushed

across a level floor at a constant speed by a force F of 600 N at an angle of 20° below the horizontal as shown in the figure.

a) What is the coefficient of kinetic friction between the crate and the floor?

mg

FNFapp

20

Ff

Fappy

Fappx

331.0

21.17051500)20(sin600

1500sin

82.563)20(cos600cos

k

N

fkNkf

N

appN

appN

appf

appf

Nkf

μ

F

FμorFμF

NF

θFF

mgFF

NθFF

FF

FμF

y

x

ExampleIf the 600 N force is instead pulling the

block at an angle of 20° above the horizontal as shown in the figure, what will be the acceleration of the crate. Assume that the coefficient of friction is the same as found in (a)

mg

FN

Ff

20

Fapp

Fappx

Fappy

2/883.0

1.15357.4288.563

1.153)20sin6001500(331.020cos600

)sin(cos

cos

sma

a

a

maθFmgμθF

maFμθF

maFF

maF

appapp

Napp

fapp

net

x

x

Inclines

cosmg

sinmg

mg

FN

Ff

Tips:Whenever there is acceleration, rotate your axes so that the acceleration is NOT angledBreak Fg into componentsWrite equations of motion or equilibriumSolve for each direction

Example – Angled Atwood’s Machine

gmamF

amgmF

maF

T

T

nety block

11

11

1

m2

m1

Masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley. As shown in the diagram, m1 is held at rest on the floor and m2 rests on a fixed incline of angle 40 degrees. The masses are released from rest, and m2 slides 1.00 m down the incline in 4 seconds. Determine (a) The acceleration of each mass (b) The coefficient of kinetic friction and (c) the tension in the string.

m1g

m2g

FNFT

FT

Ff

40

40

amFFθgm

maF

Tf

netx block

22 )(sin2

m2gcos40

m2gsin40

mass 1 accelerates only in the y-direction

mass 2 accelerates only in the x-direction (since you chose to tilt the axes)

Example – cont.

θgm

amgmamθgmμ

θgmμamgmamθgm

amgmamθgmμθgm

amgmamFμθgm

amgmamFθgm

amFFθgm

k

k

k

Nk

f

Tf

cos

sin

cossin

cossin

sin

)(sin

sin

2

2112

22112

21122

2112

2112

22

2

2

2

/125.0

)4(2101

21

sma

a

attvx ox

gmamFT 11

amFFθgm Tf 22 )(sin

NFT 5.40)10(4)125(.4

235.094.68

125.1405.085.57

kμ

Newton’s Third Law (N3L)“For every action there is an EQUAL and

OPPOSITE reaction. This law focuses on action/reaction pairs (forces) The forces NEVER cancel out, because they act on

different objects

All you do is SWITCH the wording!•PERSON on WALL•WALL on PERSON

N3LThis figure shows the force during a collision between a truck and a train. You can clearly see the forces are EQUAL and OPPOSITE. To help you understand the law better, look at this situation from the point of view of Newton’s Second Law.

TrainTrainTruckTruck

TrainTruck

aMAm

FF

There is a balance between the mass and acceleration. One object usually has a LARGE MASS and a SMALL ACCELERATION, while the other has a SMALL MASS (comparatively) and a LARGE ACCELERATION.

N3L ExamplesAction: HAMMER HITS NAILReaction: NAIL HITS HAMMER

Action: Earth pulls on YOUReaction: YOU pull on the earth

Newton’s Law of GravitationWhat causes YOU to be pulled down? THE EARTH….or

more specifically…the EARTH’S MASS. Anything that has MASS has a gravitational pull towards it.

MmFg What the proportionality above is saying is that for there to be a FORCE DUE TO GRAVITY on something there must be at least 2 masses involved.

N.L.o.G.As you move AWAY from the earth, your DISTANCE increases and your FORCE DUE TO GRAVITY decrease. This is a special INVERSE relationship called an Inverse-Square.

2

1

rFg

The “r” stands for SEPARATION DISTANCE and is the distance between the CENTERS OF MASS of the 2 objects. We us the symbol “r” as it symbolizes the radius. Gravitation is closely related to circular motion as you will discover later.

N.L.o.G – Putting it all together

221

2

227

221

1067.6

Constant nalGravitatio UniversalG

alityproportion ofconstant

r

mmGF

kgNmxG

Gr

mmF

g

g

earth eLEAVING th areyou when thisUse

earth on the areyou when thisUse

221

r

mmGF

mgF

g

g

Try this!

earth eLEAVING th areyou when thisUse

earth on the areyou when thisUse

221

r

mmGF

mgF

g

g

mxr

kgxM

r

MGg

r

MmGmg

6

24

2

2

1037.6 Earth theof radius

1097.5Earth theof Mass

Let’s set the 2 equations equal to each other since they BOTH represent your weight or force due to gravity

SOLVE FOR g!

226

2427

/81.9)1037.6(

)1097.5)(1067.6(sm

x

xxg