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

Isaac Newton (1643-1717) 2 Epitaph: "Nature and Nature's laws lay hid in night: God said, 'Let Newton be!' and all was light."

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

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

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

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.

b) Second law: Force is the cause 7 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

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

2. Fundamental Forces of Nature 9 (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.

(b) Gravity: Newton’s 4th Law 10 (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."

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

(iii) Cavendish Experiment: 1797 12 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!

(c) The Acceleration of Gravity 13 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!

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

(a) Field (Body) Forces 15 “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."

(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 1791 - 1867

(ii). Definition of Mass 17 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

(iii) Point Mass Theorem 18 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.

(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”

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)

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

(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:

(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)

Static vs Kinetic Friction 24 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:

Coefficients of Friction 25 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

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

1. Inertia & Newton’s Laws 28 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

(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)

(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:

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

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.

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

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”

(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:

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

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

1. Newton’s 3rd Law of Reciprocity 38 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.

Discuss what I found on the Web 39 Is this Newton’s 3rd Law? Reference: http://theory.uwinnipeg.ca/mod_tech/node24.html

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)

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)

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

(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:

(c) Center of Mass of System 44 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:

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: http://theory.uwinnipeg.ca/mod_tech/node24.html