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Chapters 5, 6
Force and Motion
Newtonian mechanics
• Describes motion and interaction of objects
• Applicable for speeds much slower than the speed of light
• Applicable on scales much greater than the atomic scale
• Applicable for inertial reference frames – frames that don’t accelerate themselves
Sir Isaac Newton(1643 – 1727)
Force
• What is a force?
• Colloquial understanding of a force – a push or a pull
• Forces can have different nature
• Forces are vectors
• Several forces can act on a single object at a time – they will add as vectors
Force superposition
• Forces applied to the same object are adding as vectors – superposition
• The net force – a vector sum of all the forces applied to the same object
Newton’s First Law
• If the net force on the body is zero, the body’s acceleration is zero
00 aFnet
Newton’s Second Law
• If the net force on the body is not zero, the body’s acceleration is not zero
• Acceleration of the body is directly proportional to the net force on the body
• The coefficient of proportionality is equal to the mass (the amount of substance) of the object
00 aFnet
m
Fa net
netFam
Newton’s Second Law
• SI unit of force kg*m/s2 = N (Newton)
• Newton’s Second Law can be applied to all the components separately
• To solve problems with Newton’s Second Law we need to consider a free-body diagram
• If the system consists of more than one body, only external forces acting on the system have to be considered
• Forces acting between the bodies of the system are internal and are not considered
Chapter 5Problem 6
Newton’s Third Law
• When two bodies interact with each other, they exert forces on each other
• The forces that interacting bodies exert on each other, are equal in magnitude and opposite in direction
CBBC FF
Forces of different origins
• Gravitational force
• Normal force
• Tension force
• Frictional force (friction)
• Drag force
• Spring force
Gravity force (a bit of Ch. 13)
• Any two (or more) massive bodies attract each other
• Gravitational force (Newton's law of gravitation)
• Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant
rr
mmGF ˆ
221
Gravity force at the surface of the Earth
g = 9.8 m/s2
jR
mmGr
r
mmGF
Earth
CrateEarthCrate
ˆˆ22
21
jmgjmR
GmF CrateCrate
Earth
EarthCrate
ˆˆ2
Gravity force at the surface of the Earth
• The apple is attracted by the Earth
• According to the Newton’s Third Law, the Earth should be attracted by the apple with the force of the same magnitude
jR
mmGr
r
mmGF
Earth
AppleEarthEarth
ˆˆ22
21
jm
R
mmG
aEarth
Earth
AppleEarth
Earthˆ
2
jm
m
R
Gm
Earth
Apple
Earth
Earth ˆ2
j
m
mg
Earth
Apple ˆ
Weight
• Weight (W) of a body is a force that the body exerts on a support as a result of gravity pull from the Earth
• Weight at the surface of the Earth: W = mg
• While the mass of a body is a constant, the weight may change under different circumstances
Tension force
• A weightless cord (string, rope, etc.) attached to the object can pull the object
• The force of the pull is tension ( T )
• The tension is pointing away from the body
Free-body diagrams
Chapter 5Problem 47
Normal force
• When the body presses against the surface (support), the surface deforms and pushes on the body with a normal force (FN) that is perpendicular to
the surface
• The nature of the normal force – reaction of the molecules and atoms to the deformation of material
Free-body diagrams
Free-body diagrams
Chapter 5Problem 41
Frictional force
• Friction ( f ) - resistance to the sliding attempt
• Direction of friction – opposite to the direction of attempted sliding (along the surface)
• The origin of friction – bonding between the sliding surfaces (microscopic cold-welding)
Static friction and kinetic friction
• Moving an object: static friction vs. kinetic
Friction coefficient
• Experiments show that friction is related to the magnitude of the normal force
• Coefficient of static friction μs
• Coefficient of kinetic friction μk
• Values of the friction coefficients depend on the combination of surfaces in contact and their conditions (experimentally determined)
Nss Ff max,
Nkk Ff
Free-body diagrams
Free-body diagrams
Chapter 6Problem 23
Drag force
• Fluid – a substance that can flow (gases, liquids)
• If there is a relative motion between a fluid and a body in this fluid, the body experiences a resistance (drag)
• Drag force (D)
D = ½CρAv2
• C - drag coefficient; ρ – fluid density; A – effective cross-sectional area of the body (area of a cross-section taken perpendicular to the velocity); v - speed
Terminal velocity
• When objects falls in air, the drag force points upward (resistance to motion)
• According to the Newton’s Second Law
ma = mg – D = mg – ½CρAv2
• As v grows, a decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed
½CρAvt2 = mg
Terminal velocity
Solving ½CρAvt2 = mg we obtain
AC
mgvt
2
vt = 300 km/h
vt = 10 km/h
Spring force
• Spring in the relaxed state
• Spring force (restoring force) acts to restore the relaxed state from a deformed state
Hooke’s law
• For relatively small deformations
• Spring force is proportional to the deformation and opposite in direction
• k – spring constant
• Spring force is a variable force
• Hooke’s law can be applied not to springs only, but to all elastic materials and objects
Robert Hooke(1635 – 1703)dkFs
Centripetal force
• For an object in a uniform circular motion, the centripetal acceleration is
• According to the Newton’s Second Law, a force must cause this acceleration – centripetal force
• A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed
R
vac
2
R
mvmaF cc
2
Centripetal force
• Centripetal forces may have different origins
• Gravitation can be a centripetal force• Tension can be a centripetal force• Etc.
Free-body diagram
Answers to the even-numbered problems
Chapter 5:
Problem 2. (a)1.88 N; (b) 0.684 N; (c) (1.88 N)ˆi + (0.684 N)ˆj
Answers to the even-numbered problems
Chapter 5:
Problem 10. (a)2.0N; (b) down
Answers to the even-numbered problems
Chapter 5:
Problem 22. (a) 5.5 kN; (b) 2.7 s; (c) 4.0; (d) 2.0
Answers to the even-numbered problems
Chapter 6:
Problem 2. 0.61
Answers to the even-numbered problems
Chapter 6:
Problem 32. 3.75
Answers to the even-numbered problems
Chapter 6:
Problem 36. 48km/h
Answers to the even-numbered problems
Chapter 6:
Problem 40. (a) 3.7 kN; (b) up; (c) 1.3 kN; (d) down
Answers to the even-numbered problems
Chapter 6:
Problem 104. (a)0.13 N; (b) 0.12