1. Physics Part 1 MECHANICS
Topic IV. Newton’s Laws
W. Pezzaglia
Updated: 2011Oct17

2. Isaac Newton ( ) 2
Epitaph: "Nature and Nature's laws lay hid in night: God said, 'Let Newton be!' and all was light."

3. Isaac Newton (1643-1717) Famous book the “Principia”, published 1687
Based on work done in 1666
Halley paid for the publication!

4. 4
IV. Newton’s Laws
The Force
Inertia
The Third Law

5. 5
A. Force
The First Two Laws
Fundamental Forces
Types of Forces

6. a). Newton’s First Law 6
1st Law: (adopted from Galileo’s laws of inertia),
Every body preserves in its state of being at rest or of moving “uniformly straight forward,” except insofar as it is compelled to change its state by forces impressed.
In other words:
bodies at rest tend to stay at rest, bodies in motion tend to stay in constant motion,
unless compelled to change by an outside force.

7. b) Second law: Force is the cause7
A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
In other words: Force = mass × acceleration
Force: “F” is the cause of acceleration
Acceleration “a” is the response to force
Mass: “m” is the inertia or “resistance” to force

8. c) Units of force Dimensions of Force are: SI Units: Newtons8
Dimensions of Force are:
SI Units: Newtons
CGS Units: Dyne
English: pound
1 pound =4.45 N

9. 2. Fundamental Forces of Nature9
(a) There are only 4 fundamental forces
Gravity
Electromagnetic (electricity & magnetism)*
The Weak Force (neutrinos)
The Strong force (holds nucleus together)
*All macroscopic forces we commonly use in mechanics (e.g. friction, normal force) are a manifestation of atomic electric forces.

10. (b) Gravity: Newton’s 4th Law10
(i) The apple tree story
"After dinner, the weather being warm, we went into the garden and drank tea, under the shade of some apple trees," wrote Stukeley, in the papers published in 1752 and previously available only to academics.
"He told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself."

11. (ii) The Law of Gravitation11
The mutual force between two bodies is proportional to their masses, and inversely proportional to distance.
Newton could not determine the Gravitation Constant “G”

12. (iii) Cavendish Experiment: 179712
Over 100 years later! G=6.67×10-11 Nm2/kg2
Very Small! To have 1 N of force would need 1220 kg masses 1 cm apart!

13. (c) The Acceleration of Gravity13
Combining Newton’s 2nd and 4th laws, we see that the mass of the test body cancels out!
Hence we derive Galileo’s law that all test bodies fall at the same acceleration “g”, independent of mass “m”
Hence if we measure “g”, we can determine the mass of the earth!

14. A.3 Types of Forces Body (Field) forces Inertial (Fictitious) Forces14
Body (Field) forces
Inertial (Fictitious) Forces
Contact Forces

15. (a) Field (Body) Forces15
“Action at a Distance” (no touching)
Huygens criticized: How can one believe that two distant masses attract one another when there is nothing between them? Nothing in Newton's theory explains how one mass can possible even know the other mass is there.
“actio in distans” (action at a distance), no mechanism proposed to transmit gravity
Newton himself writes: "...that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that, I believe no man, who has in philosophic matters a competent faculty of thinking, could ever fall into it."

16. (i) The field concept 16 1821 proposes ideas of “Lines of Force”
Example: iron filings over a magnetic show field lines
Gravitational Analogy:
Earth’s mass “M” creates a gravity field “g”
Force of field on mass “m” is: F=mg
(i.e. “weight”)
This eliminates “action at a distance”
Michael Faraday

17. (ii). Definition of Mass17
There are 3 ways to think about mass
Inertial Mass F=ma
Passive Gravitational Mass F=mg
Active Gravitational Mass
The “Weak Equivalence principle” says that inertial mass equals passive gravitational mass

18. (iii) Point Mass Theorem18
At what point does the force act?
The force (of gravity) acts on every piece of the mass.
Newton shows its equivalent to having all the force act at the center of the body.

19. (b) Inertial Forces 19
If one is in an accelerated reference frame (i.e. NOT moving at uniform velocity), then you experience “fictitious forces”.
Centrifugal Force
Coriolis Force
Euler Force
“Elevator Force”

20. Apparent Weight 20
Reference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment.
The “apparent” weight in an accelerating frame is: F=m(g-a)

21. A.3.c Contact Forces The “Normal” force Friction Forces21
Involves “touching”, interaction at surface
The “Normal” force
Friction Forces
Tensile Forces (ropes)

22. (i) The Normal Force (i) Normal Force “N” 22
force perpendicular to surface
Aka “force of constraint”
Since the block does not move in the direction of the normal, by Newton’s second law, the sum of the forces in that direction must be zero. Hence we can equate it to the component of weight in that direction:

23. (ii) Friction 23 Laws of Friction:
The force of friction is directly proportional to the applied load. (Amontons 1st Law, 1699)
The force of friction is independent of the apparent area of contact. (Amontons 2nd Law, 1699)
Kinetic friction is independent of the sliding velocity. (Coulomb's Law, 1781)
Note then that the Friction Force “f”:
Tangent to surface
In opposite direction to motion
Proportional to normal force (Amontons' First Law ):
“” is coefficient of friction (unitless)

24. Static vs Kinetic Friction24
Static Friction:
An undisturbed object at rest may have ZERO friction. As you push with force “F”, the friction will be exactly opposite such that the block does not move.
The maximum friction possible is given by
Hence we can say:

25. Coefficients of Friction25
Starting at rest, if we push harder and harder, the friction will increase until we get to the static maximum
Then the object will start to move, and the kinetic friction is always less

26. (iii) Tensile Forces (Ropes)26
Force is along line of the rope
The force is “transmitted” along the rope.
Pulleys can change the direction of the force.

27. 27
B. Inertia
Inertia and the first 2 laws
Statics
Dynamics

28. 1. Inertia & Newton’s Laws28
Principles restated:
A body at rest (or in uniform motion) will stay at rest (in uniform motion) unless acted upon by a NET force.
2nd Law: the VECTOR SUM of all of the forces on a body equals its mass  acceleration

29. (b) Static Equilibrium (inertia of rest)29
If system is at rest, then acceleration must be zero
Hence, vector sum of forces is zero.
Example: mass on table, the normal force “N” (up) must exactly cancel the weight “W” due to gravity (down)

30. (c) Dynamic Equilibrium [moving, but acceleration=0]30
Example: Terminal Velocity:
A falling body will accelerate due to gravity “g”
Air friction (“drag”) increases with speed (Stoke’s law)
Total force decreases with speed, hence acceleration decreases:

31. Terminal Velocity 31
Hence, body will reach a “terminal speed” when forces cancel, i.e. when body is in “dynamic equilibrium”
Note terminal velocity for skydiving is approximately 124 mi/hour or 54 m/s

32. 2. STATICS [at rest, acceleration =0]32
For equilibrium the VECTOR sum of forces is zero.
Equilibrium condition is really two equations in one.
Sum of forces in ANY direction is zero.

33. (a) Example: Hanging Object33
Write out vectors
Sum of forces in each direction gives two equation in 2 unknowns. Solve (Cramer’s rule)

34. Example: Inclined Plane What is maximum angle before it will slide?34
Trick: use coordinate system tilted by angle 
Sum of forces in each direction gives two equation in 2 unknowns. Solve (Cramer’s rule)
Solution (substitution, eliminate N) yields the critical “angle of repose”

35. (c) Drag the Block 35
For a a given angle and friction, what is the minimum force need to get block to slide? [Note, part of your tension actually lifts the block which results in less normal force, hence less friction]
Sum of forces in each direction gives two equation in 2 unknowns. Solve (Cramer’s rule)
Solution (substitution, eliminate N) yields:

36. 3. Dynamics [accelerating systems]36
Rope climbing problem
Inclined Plane (Galileo’s law)
Inclined plane with friction
Details done in class (see lecture notes)

37. C. Third Law and Systems The Third Law Compound Systems37
C. Third Law and Systems
The Third Law
Compound Systems
Point Mass Theorem

38. 1. Newton’s 3rd Law of Reciprocity38
3rd Law (1687)
“To any action there is always an opposite and equal reaction:
or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts”
This means, as much as the earth is pulling on you
You are pulling back on the earth the same amount
Without this law, energy and momentum would not be conserved in the universe.

39. Discuss what I found on the Web39
Is this Newton’s 3rd Law?
Reference:

40. 2. Compound Systems Details done in class (see lecture notes)40
Hanging Weights, Train Problems
Atwood’s machine (1784)
More Exotic problems
Details done in class (see lecture notes)

41. 3. Extended Bodies Details done in class (see lecture notes)41
Internal vs External Forces
Euler’s Equation of Motion (1750)
Center of Mass
Details done in class (see lecture notes)

42. (a) Internal vs. External Force42
Sum over all forces in system.
Internal forces cancel due to Newton 3rd law
If all parts have same acceleration, then,

43. (b) Euler’s Equations of Motion (1750)43
For a complex system, different parts might have different accelerations.
Euler shows that there is a “center of mass” of the system for which we can imagine all the mass is concentrated, and the motion of this center follows Newton’s 2nd law:

44. (c) Center of Mass of System44
Concept introduced by Archimedes of Syracuse
Pappus of Alexandria extended the idea to calculate “centroids” of geometric objects.
Euler extends ideas over Newton’s laws, shows:

45. References and Notes 45
Statics and hanging by 2 ropes, see “Lami’s Theorem”. Also considered by Leonardo da Vinci
The Principia, I. Newton, translated by Cohen and Whitman, 1999, University of California Press.
Wrong interpretation of Newton’s 3rd law at: