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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
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