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