Chapters 4, 5

Force and Laws of Motion

What causes motion?

• That’s the wrong question!

• 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

Aristotle(384 BC – 322 BC)

Galileo Galilei(1564 – 1642)

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

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

2112 FF

Forces of different origins

• Gravitational force

• Normal force

• Tension force

• Frictional force (friction)

• Drag force

• Spring force

Gravity force (a bit of Ch. 8)

• 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 4Problem 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/s2. 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 equal to the gravitational force of the object

Free-body diagrams

Free-body diagrams

Chapter 5Problem 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

• 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

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)

nf ss max,

nf kk

Free-body diagrams

Free-body diagrams

Chapter 5Problem 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

• 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 (R)

R = ½DρAv2

• 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)

• According to the Newton’s Second Law

ma = mg – R = mg – ½DρAv2

• As v grows, a decreases. At some point acceleration becomes zero, and the speed value riches maximum value – terminal speed

½DρAvt2 = mg

Terminal velocity

• Solving ½DρAvt2 = mg we obtain

AD

mgvt

2

vt = 300 km/h

vt = 10 km/h

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.

Centripetal force

• Centripetal forces may have different origins

• Gravitation can be a centripetal force• Tension can be a centripetal force• Etc.

Free-body diagram

Chapter 5Problem 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 207.7 cm

Answers to the even-numbered problems

Chapter 4

Problem 26590 N

Answers to the even-numbered problems

Chapter 4

Problem 385.77 N; 72.3°

Answers to the even-numbered problems

Chapter 5

Problem 28580 N; opposite to the motion of the

cabinet

Answers to the even-numbered problems

Chapter 5

Problem 50110 m