Transcript Chapters 4, 5 Force and Laws of Motion
Chapters 4, 5 Force and Laws of Motion
What causes motion?
Aristotle (384 BC – 322 BC) •
That’s the wrong question!
Galileo Galilei (1564 – 1642) •
The ancient Greek philosopher Aristotle believed that forces - pushes and pulls - caused motion
•
The Aristotelian view prevailed for some 2000 years
•
Galileo first discovered the correct relation between force and motion
•
Force causes not motion itself but change in motion
Newtonian mechanics
Sir Isaac Newton (1643 – 1727) •
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
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Applicable for inertial that don’t accelerate reference frames themselves – frames
Force
•
What is a force?
•
Colloquial understanding of a force – a push or a pull
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Forces can have different nature
•
Forces are vectors
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Several forces can act on a single object at a time – they will add as vectors
Force superposition
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Forces applied to the same object vectors – superposition are adding as
•
The net force – a vector sum of to the same object all the forces applied
Newton’s First Law
•
If the net force on the body is zero, the body’s acceleration is zero
F net
0
a
0
Newton’s Second Law
•
If the net force on the body is not zero, the body’s acceleration is not zero
F net
0
a
0 •
Acceleration of the body is directly proportional to the net force on the body
•
The coefficient of proportionality mass (the amount of substance) of the object
m a
F net
a
is equal to the
F net m
Newton’s Second Law
•
SI unit of force kg*m/s 2 = N ( Newton )
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Newton’s Second Law can be applied to all the components separately
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To solve problems with Newton’s Second Law we need to consider a free-body diagram
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If the system consists of more than one body, only external forces acting on the system have to be considered
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Forces acting between the bodies of the system are internal and are not considered
Newton’s Third Law
•
When two bodies interact forces on each other with each other, they exert
•
The forces that interacting bodies exert on each other, are equal in magnitude and opposite in direction
F
12
F
21
Forces of different origins
•
Gravitational force
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Normal force
•
Tension force
•
Frictional force (friction)
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Drag force
•
Spring force
Gravity force (a bit of Ch. 8)
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Any two (or more) massive bodies attract each other
•
Gravitational force (Newton's law of gravitation)
F
G m
1
m
2
r
2
r
ˆ •
Gravitational constant 6.67*10 –11 m 3 /(kg*s 2 ) G = 6.67*10 –11 N*m – universal constant 2 /kg 2 =
Gravity force at the surface of the Earth
F Crate
G m
1
m
2
r
2
r
ˆ
G m Earth m Crate
2
R Earth j
ˆ
F Crate
Gm Earth
2
R Earth
m Crate j
ˆ
m Crate j
ˆ
g
= 9.8 m/s 2
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
F Earth
G m
1
m
2
r
2
r
ˆ
G m Earth m Apple
2
R Earth j
ˆ
a Earth
G m Earth m Apple
2
R Earth m Earth j
ˆ
Gm Earth
2
R Earth
m Apple m Earth j
ˆ
m m Apple Earth j
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
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A weightless cord (string, rope, etc.) attached to the object can pull the object
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The force of the pull is tension (
T
)
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The tension is pointing away from the body
Free-body diagrams
Chapter 4 Problem 56 Your engineering firm is asked to specify the maximum load for the elevators in a new building. Each elevator has mass 490 kg when empty and maximum acceleration 2.24 m/s 2 . The elevator cables can withstand a maximum tension of 19.5 kN before breaking. For safety, you need to ensure that the tension never exceeds two-thirds of that value. What do you specify for the maximum load? How many 70-kg people is that?
Normal force
•
When the body presses against the surface (support), the surface deforms and pushes on the body with a normal force (
n
) that is perpendicular to the surface
•
The nature of the normal force – reaction of the molecules and atoms to the deformation of material
Normal force
•
The normal force is not always gravitational force of the object equal to the
Free-body diagrams
Free-body diagrams
Chapter 5 Problem 19 If the left-hand slope in the figure makes a 60 ° angle with the horizontal, and the right-hand slope makes a 20 ° angle, how should the masses compare if the objects are not to slide along the frictionless slopes?
Spring force
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Spring in the relaxed state
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Spring force (restoring force) acts to restore the relaxed state from a deformed state
Hooke’s law
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For relatively small deformations
F s
k d
Robert Hooke (1635 – 1703) •
Spring force is proportional to the deformation and opposite in direction
•
k
– spring constant
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Spring force is a variable force
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Hooke’s law can be applied not to springs only, but to all elastic materials and objects
Frictional force
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Friction (
f
) - resistance to the sliding attempt
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Direction of friction – opposite to the direction of attempted sliding (along the surface)
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The origin of friction – bonding between the sliding surfaces (microscopic cold-welding )
Static friction and kinetic friction
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Moving an object: static friction vs. kinetic
Friction coefficient
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Experiments show that friction is related to the magnitude of the normal force
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Coefficient of static friction
μ s f s
, max
s n
•
Coefficient of kinetic friction
μ k f k
k n
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Values of the friction coefficients depend on the combination of surfaces in contact and their conditions (experimentally determined)
Free-body diagrams
Free-body diagrams
Chapter 5 Problem 30 Starting from rest, a skier slides 100 m down a 28 ° slope. How much longer does the run take if the coefficient of kinetic friction is 0.17 instead of 0?
Drag force
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Fluid – a substance that can flow (gases, liquids)
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If there is a relative motion between a fluid and a body in this fluid, the body experiences a resistance (drag)
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Drag force (
R
)
R = ½DρAv 2
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D
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)
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According to the Newton’s Second Law
ma = mg – R = mg – ½DρAv 2
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As
v
grows,
a
decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed
½DρAv t 2 = mg
Terminal velocity
•
Solving
½DρAv t 2 = mg
we obtain
v t
2
mg D
A
v t = 300 km/h v t = 10 km/h
Centripetal force
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For an object in a uniform circular motion, the centripetal acceleration is
a c
v
2
R
•
According to the Newton’s Second Law, a force must cause this acceleration – centripetal force
F c
ma c
mv
2
R
•
A centripetal force accelerates a body by changing the direction of the body’s velocity without changing the speed
Centripetal force
•
Centripetal forces may have different origins
• • •
Gravitation can be a centripetal force Tension can be a centripetal force Etc.
Centripetal force
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Centripetal forces may have different origins
• • •
Gravitation can be a centripetal force Tension can be a centripetal force Etc.
Free-body diagram
Chapter 5 Problem 25 You’re investigating a subway accident in which a train derailed while rounding an unbanked curve of radius 132 m, and you’re asked to estimate whether the train exceeded the 45-km/h speed limit for this curve. You interview a passenger who had been standing and holding onto a strap; she noticed that an unused strap was hanging at about a 15 ° angle to the vertical just before the accident. What do you conclude?
Answers to the even-numbered problems Chapter 4 Problem 20 7.7 cm
Answers to the even-numbered problems Chapter 4 Problem 26 590 N
Answers to the even-numbered problems Chapter 4 Problem 38 5.77 N; 72.3
°
Answers to the even-numbered problems Chapter 5 Problem 28 580 N; opposite to the motion of the cabinet
Answers to the even-numbered problems Chapter 5 Problem 50 110 m