Physics 61: Autumn 2008
Mechanics and Special Relativity

Preliminaries

General Information

Location and Time

201 Hewlett, Tu, Thu 9.00 - 10.50 a

Lecturer

Roger Blandford (rdb3@stanford.edu, 207 Physics–Astrophysics Building). Office Hours: 12-1p Tuesdays

Admin

Christine Aguilar (Christine.aguilar@stanford.edu, 206 Physics–Astrophysics Building)

TAs

Aurelien Bouvier (bouvier@stanford.edu)
Leyan Lo (leyanlo@stanford.edu)
Kevin O’Sullivan (kosok@stanford.edu)
Slavosh Rezvan (siavoshr@stanford.edu)

Sections will meet weekly.

Course goals

(i) to solidify understanding of Newtonian mechanics and energy
(ii) to provide a basis for the study of electromagnetism (Ph 63) and thermodynamics plus quantum mechanics (Ph65)
(iii) to prepare you for the study of Lagrangian mechanics
(iv) to understand fundamentals of special relativity with confidence
(v) to develop confidence in handling vectors, matrices and differential equations.

Recommended Preparation

The equivalent of AP Physics C and AP Calculus B/C with 5 on one and at least 4 on the other. Students with a good background in physics and calculus below this level should start with Physics 45. Students with less preparation should start with Physics 41 in the Winter quarter. It is entirely possible to catch up and become a Physics major starting with Physics 41 or Ph 45. Do what is right for you and consult the TAs or me.

Lectures

In addition to discussing material on the synopsis, I will show some demonstrations and present some “Historical Asides” or HA and “Points to Ponder” or P2P. These will not be examined. The P2P are intended to start some of you thinking about questions that will be answered later in this course or in subsequent courses. I strongly advise that you not try to read ahead in more advanced texts for these topics but instead put yourself in the position of the great dynamicists of the past and think about them by yourself.

Discussion Sections

This year the TAs will be leading discussion sections designed to promote group and active learning. We are all trying to understand this and so please be patient and give us lots of feedback.

Text

Required

David Morin - Introduction to Classical Mechanics (Cambridge)
New text with many exercises!
Will cover roughly 3/4 omitting Lagrangians, some of the rotational dynamics and general relativity

Auxiliary

Kleppner & Kolenkow – An Introduction to Mechanics
Taylor & Wheeler – Spacetime Physics
Taylor – Classical Mechanics

Readings

I will assign various readings from Morin. The first one is the general advice for solving problems - sketch, knowns->unknowns, algebra, dimensions, limits, order of magnitude - which I shall illustrate throughout the course.

Grading

Homeworks

Issued Thursday and due at the start of class the following Thursday
There will be eight homeworks. Only the first seven will be graded out of 5 percent  each and we will drop your lowest score. 
The best test of whether or not you have mastered the material is if you can calculate with accuracy and confidence. This requires practice. All of the questions will be drawn from Morin or written out in full. Another good test of understanding is to explain the material to a non-physicist. I will start each homework with a short exercise to give you practice at doing this. I will finish each homework with a “Question to Ponder” or Q2P. These are similar to the P2P and are not required and will not count towards your grade. Try to think about these questions without consulting other sources, if you have time.
You are encouraged to join study groups to work on homeworks. However, you must write up your answers independently. You are required to abide by the Stanford Honor Code.
(http://www. stanford.edu/dept/vpsa/judicialaffairs/guiding/honorcode.int.htm).

Midterm

2hr in class exam on 10/21 (20 percent)

Final

3 hr exam,  tentatively on 12/9 (50 percent)

GOOD LUCK!

Lecture 1 Kinematics (23/9/07)

Goals

--To illustrate some general techniques for solving problens.
--To introduce basic kinematic elements of mechanics - space, time, point, speed, acceleration
--To review application of differential and integral calculus to description of motion
--To review Taylor series and elementary functions
--To solve some exercise

Reading

M 1-11, 675-678

Elements

Space

Greeks systematized geometry and reduced it to axioms (Euclid)These rules led to notions of distance  angle, parallelism etc. They also imply Pythagorus' theorem. These are statements about measurement in a background space, but what is space?
Newton said “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable…”.
Various philosophers weighed in, notably Kant and regarded Euclidean space as an a priori truth. We now know that  
(i) This is wrong mathematically and physically. We can invent quite different geometries; space in the presence of gravitation is curved in the sense that Pythagorus' theorem, is untrue - eg on sphere.
(ii) Knowing how badly it fails allows us to quantify how accurate Euclidean geometry is for most everyday applications. Bottom line is that it is perfectly adequate! Einstein validates not invalidates Newton in practice. (Same is true for quantum mechanics.) Euclidean geometry implicit henceforth in this course.
(iii) There could be more than three space dimensions! We will ignore this possiblity.
Unit of length is meter - originally arm span...  (10^(-7)) of quadrant passing through Paris, then became a Pt bar, then wavelength of Kr86 or He-Ne spectral line.

Time

According to Newton: “Absolute true, and mathematical time, of itself and from its own nature flows equably without relation to anything external…’’  but what is it?
Philosophers and poets, artists and authors have their own perspectives which are valuable, but we are physicists!
Two philosophical views
- Realists (Platonists).  There exists an objective framework independent of us and we explore it
- Positivists. Realist view is irrelevant. What matters is relationships between our sensory perceptions or measurements.
These matters are occasionally important for Newtonian dynamics in practice, somewhat important for special relativity and very important for understanding quantum mechanics. We shall not make a big deal about here but I will point out when it appears.
Unit of time is second - originally one's pulse then pendula and classical oscillators (eg Harrison's chronometer used for longitude). Also used astronomical definitions eclipses, pulsars etc. Today use atomic clocks based upon Cs137 which are accurate to better than a s in (10^15)s.

Spacetime

Newton saw space and time as "containers" in which events were located but space and time were quite separate. That is the way we will develop classical mechanics. However, Einstein unified them as  spacetime and our discussion of relativity will be based on that. Relativity is built around a special speed, that of light,c=3 x (10^8)m (s^(-1)). An immediate practical consequence is that the meaurement of our unit of length is c x our unit of time.

Point particle

A point particle is a mathematical abstraction useful for small bodies. Usually need to include internal motions of the body.

Point to Ponder (P2P)

You are used to three spatial dimensions; there may be more. How do feel about two (or more) temporal dimensions?

Kinematics

Motion in 1D

Average speed v=Dx/Δt. Take the limit to define instantaneous speed,  
v=dx/dt≡(OverDot[x])⇔x=∫dt v. This is the fundamental theorem of calculus!
Examples: speed of light c = 3 x (10^8)m (s^(-1)), speed of sun in Galaxy ~ (10^(-3))c,  speed of Earth ~ (10^(-4))c, speed of sound ~ (10^(-6))c, running ~(10^(-8))c. Good for order of magnitude estimates
Average acceleration a=Dv/Δt ->dv/dt in limit, v=∫dt a etc. For const. a x=(Subscript[v, 0])t+(at^2)/2, v=(Subscript[v, 0])+at=>(v^2)=((Subscript[v, 0])^2)+2ax etc 
Examples: g=10 m (s^(-2))(actually 9.8) constant. Drop a ball from rest for 1 s, v=10 m (s^(-1)), <v>= 5 m (s^(-1)), x=5 m. Bounce -m bring to rest in ~ 1 mm or ~ 100 μs, a ~-(10^4)g. (OM) 
Accelerate at g for a year (~30 Ms) => v~c reach distant galaxies!

Taylor series

This is an example of a Taylor series [M 675-6], f(a+x) =f(a)+xf'(a)+(x^2)f''(a)/2+(x^3)f'''(a)/6+... Practice this for simple functions.

Taylor=Normal[Series[((Sin[x])/x),{x,0,20}]]
Plot[{((Sin[x])/x),Taylor},{x,0,10}]

Always check convergence.

Progress in Physics

Accelerating universe

The rate of change of acceleration is sometimes called the jerk. We have discovered that the universe is now accelerating in the separation r of any two distant galaxies is now accelerating while it was accelerating in the past. There is positive jerk j=(OverDot[a]). It turns out that r=A (sinh^(2/3))Bt, where the constants A, B depend upon the iniital conditions. Let us evaluate j.

r[t_]:=A ((Sinh[B t])^(2/3))
Plot[r[t]/.{A->1,B->1},{t,0,2},Frame->True]
v[t_]:=D[r[t],t]
a[t_]:=D[v[t],t]
j[t_]:=D[a[t],t]
Simplify[v[t]]
Simplify[a[t]]
Simplify[j[t]]
Simplify[((v[t])^3)/((r[t])^2)]

Hence j=(v^3)(r^(-2)). Note that it is dimensionally homogeneous. This is a remarkable relation!

Historical Aside HA

Galileo is credited with first understanding kinematics and realizing that acceleration under gravitational acceleration was independent of mass. Originally he was a scholastic and used clever though wrong arguments to argue that v ~ x when a body is dropped.  He then famously used experiments to discover that v ~ t, Amusingly, at late times in the universe  j~a~v~x~Exp[Ct] for some constant C and v~r, so scholastic arguments may be good for something!

HA

The dot notation for differentiation is due to Newton, the d/dt to Leibnitz. They fought!

Problems

M 1.4, 1.5, 1.6

Summary

--Is this course for you?

--Problem solving techniques: sketch, knowns->unknowns, algebra, dimensions, limits, order of magnitude.

--Kinematics: elements of the description of motion in 1D; position, speed, acceleration, jerk...

--Differential and integral calculus. Need facility to solve problems with confidence. If you don't have this, practice!

--Taylor series. General technique, elementary functions, convergence, practice again!

Lecture 2 Vectors (25/9/07)

Goals

--To introduce vectors as geometrical objects
--Length and unit vectors
--Scalar vector and triple products
--Differentiating and integrating vectors wrt t.

Reading

M679 - 682, M68-70.

Φ[x_,y_]==-3((((((x+1.5))^2)+(y^2)+1))^(-.5))-8((((((x-1.5))^2)+(y^2)+1))^(-.5))-20((((x^2)+(((y-1.5 (3^(.5))))^2)+1))^(-.5))
ContourPlot[Φ[x,y],{x,-2,2},{y,-1,3}]

Vectors in physics

1 D -> 3 D

So far we just worked in 1D, However we live and calculate in 3D. In 1D we implicitly specify the position of a particle by asking how many units of length (armspans, Pt bars, wavelengths etc) there are from a specific, though arbitrary origin and the particle. We can do the same in 3D using three real numbers. The origin if not where "I" am standing is at least somewhere I choose - the positivist approach. Furthermore, the numbers that I write down depend upon the orientation of the axes that I choose. Although this is sometimes natural, it is essentially arbitrary.

Vectors as geometrical objects

This is the realist approach. Vectors have an existence in space that is independent of our choice of coordinate system. We do not need three numbers to describe them. One symbol will do. The most elementary vector is the displacement vector that connects one place to another. By convention, it "lives" here. We will call it r.  The here is usually specific. It might be the vertex or center of mass of a solid body. It might be where a particle is now with the there being where it will be in one second. If it is not specific, it does not matter.

Vector spaces

Vectors have magnitude (length) and direction. If we want "there" to be twice as far away, we regard the vector as being twice as long or multiplied by a factor 2. In other words we can multiply vectors by real numbers or scalars. Likewise if we add a vector from here to there to one from there to there again, we consider this as a displacement from here to there again. We have a rule of adding vectors. This physical description therefore has the same features as the more abstract mathematical definition of a vector space. Furthermore the mathematical properties of distribution, association, commutation, inverse and identity are also properties of physical vectors.

Length and unit vectors

Given a unit of physical distance, we can define the length of a vector to be the number of these units between here and there. Notationally, this is r=|r|, a postive number. A unit vector can be defined by (OverHat[r])=r/r. We can use the techniques of Euclidean geometry to calculate the length and direction of the sum of two vectors.

Laws of Physics

We believe that the fundamental laws of physics that ultimately govern the outcome of an experiment do not depend upon the location or orientation of the laboratory. Of course playing golf is different on the Earth and the moon, but the fundamental laws are the same. For this reason, we argue that the fundamental laws can be expressed as relationships between geometrical objects like vectors and do not depend upon a coordinate system. In practice, this strongly constrains the form a law of physics can have.

P2P

However, there are special places and directions  in the universe, the center of the Earth, the sun, the center of the Galaxy... Why can't an engineer, say, acknowledge this and always set up problems calling out special places and directions? What happens in practice? What does this tell you about the difference between engineers and physicists?

Vectors in mechanics

Let us generalize our treatment of kinematics to 3D. The motion of a particle instead of being described as x(t) is now r(t). t is an independent variable. If the particle moves in a straight line, v(t)=((r ((Subscript[t, 2]))-r ((Subscript[t, 1])))/((Subscript[t, 2])-(Subscript[t, 1]))). (Note that we are adding one vector to the inverse of another and then multiplying by a scalar in the language of mathematics.) Taking the limit, v=dr/dt and so on for a, j etc. Similarly, we can treat an integral as the sum of many vectors multiplied by a scalar and then taking the limit, e.g. r(t)=r(0)+((∫_0)^t)dt'v(t').

Scalar (dot) product

We define the scalar product of a vector with itself as the square of its length; v·v=(v^2). The scalar product of different vectors is given by v·w=[(((v+w))^2)-(((v-w))^2)]/4 =vw cos ϕ, where ϕ is the angle between the two vectors as is easy to demonstrate. Note that we are converting two vectors into a scalar. One use of scalar products is in defining work ∫dr·F.

Cartesian coordinates

So far, this has had nothing to do with coordinates. If we choose a fixed coordinate system, then we define unit vectors along the coordinate axes, i,j,k, say, and we can write a vector v as (Subscript[v, 1])i+(Subscript[v, 2])j+(Subscript[v, 3])ketc. Mostly we will work with a Cartesian basis where  i,j,k are mutually perpendicular and i·j=0; i·i=1 etc. We call this orthonormal.  In this case, we write i=(OverHat[x]) etc. The length of v is v=(((((Subscript[v, x])^2)+((Subscript[v, y])^2)+((Subscript[v, z])^2)))^(1/2)). Note that the scalar product is v·w=(Subscript[v, x])(Subscript[w, x])+(Subscript[v, y])(Subscript[w, y])+(Subscript[v, z])(Subscript[w, z]).

Vector (cross) product

There is a second product of two vectors that produces a vector result. We expect that v x w ∝ vw, but where does it point? If it lies in the plane defined by the two arguments, then we cannot make  v x w=-v x -w. However if it is perpendicular to this plane, then we can have a consistent definition as long as it is anti-commutative v x w=-w x v which implies that v x v=0. Now let us consider orthonormal unit vectors (OverHat[x]),(OverHat[y]),(OverHat[z]). A clear choice with the right properties is (OverHat[x])  x  (OverHat[y]) =(OverHat[z]) and so on cyclically. It is convenient to write this as a determinant.
v x w = |({{(OverHat[x]), (OverHat[y]) , (OverHat[z])}, {(Subscript[v, x]), (Subscript[v, y]), (Subscript[v, z])}, {(Subscript[w, x]), (Subscript[w, y]), (Subscript[w, z])}})| from which we find that (v x w) · (v x w)=(v^2)(w^2)-v · (w^2)=>|v x w|=v w sinϕ where . This is a plausibility argument about the mathematics.It turns out to be very useful. Note that the sign of a cross product changes if we look at it in a mirror, unlike a dot product. 
One example of a vector product  is an element of area dA = da x db.
Another introduces the idea of the angular velocity vector ω . Consider a point orbiting the origin with radius r. Let it rotate through dϕ in dt. The angular speed is ω=dϕ/dt, which is all you need in 2D. However, in 3D the plane of rotation is arbitrary and we can specify it using the convention of directing the angular velocity perpendicular to the plane. We then find that for constant ω,
v = ω x r;  a = ω x v = ω x (ω x r)= (ω · r)ω-(ω^2)r

Triple Product

Another scalar can be formed from three vectors. u·(v x w)=(u x v) ·w =-v·(u x w) etc.

u  · v x w = |({{(Subscript[u, x]), (Subscript[u, y]) , (Subscript[u, z])}, {(Subscript[v, x]), (Subscript[v, y]), (Subscript[v, z])}, {(Subscript[w, x]), (Subscript[w, y]), (Subscript[w, z])}})|. It is useful for volumes. dV=da · ( db x dc) etc

Polar coordinates

Most of what we will do will use Cartesian coordinates simply because this is quickest. However, our physical answers should not depend on this choice and sometimes another coordinate system is preferable. A good example is polar coordinates (r,θ). The unit vectors are (OverHat[r]) = cosθ (OverHat[x])+ sinθ (OverHat[y]) ;  (OverHat[ θ]) =- sinθ (OverHat[x])+ cosθ (OverHat[y]) . So d (OverHat[r]) /dt=(OverDot[θ])(OverHat[θ]), d(OverHat[ θ]) /dt=-(OverDot[θ])(OverHat[r]) and                              
 v = (OverDot[r]) (OverHat[r ])+ r (OverDot[θ]) (OverHat[θ]); a = ((Overscript[r, ..]) - r((OverDot[θ])^2))(OverHat[r]) + (r (Overscript[θ, ..]) +2 (OverDot[r ])(OverDot[θ]))(OverHat[θ]) . We will use this later
 M3.22, 3.23

HA Josiah Willard Gibbs

Gibbs was a professor of physics at Yale from 1869-1903 who pioneered much of modern thermodynamics and also (independently of an Englishman called Heaviside) developed modern vector notation.

Summary

Physical vectors

Relation to vector spaces

Useful formulas

r(t)=r(0)+((∫_0)^t)dt'v(t'), v=(((((Subscript[v, x])^2)+((Subscript[v, y])^2)+((Subscript[v, z])^2)))^(1/2)); v·w = vw cosϕ; v x w = |({{(OverHat[x]), (OverHat[y]) , (OverHat[z])}, {(Subscript[v, x]), (Subscript[v, y]), (Subscript[v, z])}, {(Subscript[w, x]), (Subscript[w, y]), (Subscript[w, z])}})| ; |v x w| = vw sinϕ

Polar coordinates

v = (OverDot[r]) (OverHat[r ])+ r (OverDot[θ]) (OverHat[θ]); a = ((Overscript[r, ..]) - r ((OverDot[θ])^2))(OverHat[r]) + (r (Overscript[θ, ..]) +2 (OverDot[r ])(OverDot[θ]))(OverHat[θ]) .

Homework 1 (due 2/10 before class)

Remember to try these problems by yourself first before the sections and study groups meet. You must write up your solutions completely independently.

1. Imagine you get to meet Aristotle and discuss dynamics. Explain carefully what is meant by speed and acceleration and why you think that the speed of a dropped stone increases linearly with the time. (Two short paragraphs will suffice.)

2. M 1.9

3. M 1.10

4. M 1.15

5. Explain why (A x B) ·C=A·(B x C)=B·(C x A)=-(A x C)·B.

6. Evaluate (A x B) x C and (A x B)·(A x B).

7. We can define the hyperbolic sine function, sinh x in terms of the exponential function by:  
    sinh x=((e^x)-(e^(-x)))/2. 
    We can also define cosh x by:  
    cosh x=(d/dx) sinh x.     
    a. Sketch a graph of both functions
    b. Evaluate (cosh^(2 ))x-(sinh^2)x  
    c. Differentiate cosh x
    d. Expand  sinh 2x, cosh 2x in terms of sinh x, cosh x.
    e. Make power series expansions to three terms of sinh x  about x=0 
    f. Sketch the function tanh x=sinh x / cosh x

8. As we discussed in class, the radial distance of a distant galaxy r(t) changes with time t since the big bang according to the differential equation(Overscript[ r, ...])=((OverDot[r])^3)/(r^2) . Verify that  r=(Subscript[r, 0])(sinh^(2/3))t   is a solution of this differential equation, where (Subscript[r, 0]) is a constant which depends upon which galaxy we choose. Sketch this function up to the present time using the fact that t is measured in units of 11 Gyr and the universe is 14 Gyr old today. How did the universe expand when it was very young? What will happen in the distant future? 
[Hint. There are several ways to show this, but if you want to calculate the answer directly then you should first show that 
(Overscript[r, ..]) = (Subscript[r, 0]) ((2(2 (cosh^2) t - 3))/(9(sinh^(4/3))t))]

Question to Ponder (Q2P1. Not for credit.) Sketch the function f(x)=exp[-(x^(-2))]  . Make a Taylor series expansion about the origin. What is going on?

Lecture 3 Newton's Laws (30/9/07)

Goals

--To understand the meaning of Newton's laws of motion and gravitation
--To understand about inertial and gravitational mass, force
--To see vectors in action

Reading

M51-59, M152, M23-35

The Laws

N1

"In the absence of forces, a body moves with constant velocity."
What does this mean?
-- Applies to isolated, slow (v<<c) macroscopic (x>>h/mv)  bodies - no relativity, quantum mechanics -  initially in the absence of gravitation, electricity, magnetism (eg in interplanetary space).
Ignore internal motions wtc.
-- It is an experimental result. Aristotle asserted that an isolated body would come to rest. The world could have been like this, but isn't!
-- It refers to velocity not speed. It is a statement about motion in 3D.
-- There is an implicit assumption, that we are in an "inertial" frame. What it is really saying is that there exist inertial frames in which it is true. Set up one coordinate system in which the position is x. Now set up a coordinate system moving with constant velocity u relative to this frame. The coordinate at (universal) time t, relative to when they are coincident is x'(t) = x(t) - u t and if x = v t, x' =(v-u) t =>v' = v-u which is also constant. Inertial frames are a family. They are key to understanding special relativity. Same inertial frames work ∀ bodies.
-- This law fails when we measure velocity in a frame that is rotating wrt inertial frame. Rotating frames are non-inertial. 
-- As an example consider a body moving with constant radial speed (OverDot[r]) with θ const. in some frame. The acceleration is 
a = (OverDot[v]) = ((Overscript[r, ..]) - r ((OverDot[θ])^2))(OverHat[r]) + (r (Overscript[θ, ..]) +2 (OverDot[r ])(OverDot[θ]))(OverHat[θ]) =0.  Now consider this in a frame that rotates with constant angular velocity ω = (OverDot[θ]). r is unchanged but now there is a finite (OverDot[θ]) and a' = -r (ω^2)(OverHat[r]) + 2 (OverDot[r ])ω(OverHat[θ]) ≠0.
-- In interplanetary space, inertial frames do not rotate with respect to distant galaxies, microwave background etc.
--P2P Why is this so?
-- Non-isolated bodies that are pushed or pulled by forces will change their velocities
--Types of macroscopic force include normal force, tension, friction, viscosity. Often approximate maximum static friction as μ  x normal force 
-- We can include gravity, electricity and magnetism by saying they exert invisible forces that are responsible for departures from this law.
-- We know what to do when v ~ c - later-  and on subatomic scale (Ph65). Instead of invalidating N's laws, these validate them by allowing us to know just how accurate they are for most everyday circumstances.

N2

"For a body of mass m the force is F = m a."
What does this mean?
-- (Inertial) mass. Consider one substance - eg water - of constant density. Double the volume, double the mass etc This is a definition of relative mass. Fix unit eg by defining the mass of a litre ((10^(-3))(m^3)) of water as 1 kg. Now use a Pt-Ir alloy.
-- It is found that the acceleration measured for the same force is inversely proportional to the mass. This is a non-trivial statement.
-- Illustrate with astronaut tug of war.
-- Masses for other substances in inverse proportion to accleration by same force.
-- Force is a vector as acceleration is a vector and acceleration parallel to force.
-- Unit of force is the Newton (SI system not Gaussian) gives 1 kg a=1 m (s^(-2)). 
--.... μ,m,0,k,M...
-- As forces are vectors they add as vectors and can participate in dot and cross products. 
-- Can now define gravitational force eg exerted by earth. Galileo found that acceleration independent of mass. Hence F = m g called weight. g =9.8(approx 10) m (s^(-2)).
-- This m is (passive) gravitational mass equal to inertial mass. Tested experimentally on Eotvos experiment. to (10^(-12))
-- P2P This is the Principle of Equivalence. It is a remarkable observation which (like the inverse square law) finds an elegant interpretation in the general theory of relativity. Try thinking about some of the questions Einstein thought. What is physics like in an accelerating elevator? What is an inertial frame when we include gravity? etc.
--EM in Ph 63

N3

"If first body exerts force on second body, second body exerts equal and opposite (reaction) force on first body."
What does this mean?
-- Experimentally true for common macroscopic forces including gravity; actually can be violated in EM - later.
-- m a equal and opposite on the two bodies
-- Allows us to deal with composite bodies. Break down into small subunits with external and internal forces (Subscript[ΣF, i])+(Subscript[ΣF, ij])=(Subscript[Σm, i])(Overscript[(Subscript[r, i]), ..]) 
-- =>defn of CM R = (Subscript[Σm, i])(Subscript[r, i])/(Subscript[Σm, i]) => F=M (Overscript[R, ..])
-- Also extend to fluids a = -(1/ρ)(dP/dx), where P is pressure
-- => gravitational force ∝ (Subscript[m, 1])(Subscript[m, 2]) =>active =passive gravitational mass

NG

" Gravitational force between two bodies varies as inverse square of their separation."
Clear what it means!
-- Empirical observation - apple has a = ((Overscript[r, ..]) - r((OverDot[θ])^2))(OverHat[r]) + (r (Overscript[θ, ..]) +2 (OverDot[r ])(OverDot[θ]))(OverHat[θ])  =>a= (Overscript[r, ..]); moon has a = r((OverDot[θ])^2) ∝ (r^(-2))
-- F=-((G (Subscript[m, E])m r)/(r^3))= m a => a = -((G (Subscript[m, E]) r)/(r^3))
--Will return to this topic!
--Cavendish experiment. 2 x 1.5 kg spheres 0.1 m apart, 2 x 0.015 kg spheres 0,02 m away. Evaluate force, torque.

Gravitational force due to spherical shell of matter

-- Preliminaries: Treat element of area as a vector of length equal to the area and directed outward dA=da x db etc. Solid angle subtended by object is fraction of sky it covers divided by 4 π. Unit is steradian. dΩ=r·dA/(r^3). Use this to prove famous result that the gravitational force exerted by a spherically symmetric distribution of matter at a point P is the same as that the would be exerted if all the mass interior to P were concentrated at the center of the distribution.

HA : Henry Cavendish (1731 - 1810)

Painfully shy man related to Duke and Duchess of Devonshire. Did not publish much of what he discovered including Hydrogen, Argon, Ohm's, Dalton's and Coulomb's Law. Cavendish Experiment actually devised by John Michell(1724-93). Cavendish did a superbly  careful experiment for the time,  got the calculation wrong but after being corrected gave the mass and density of Earth to better than two significant figures!

HA: Galileo and Newton

Galileo

1564-1642 experiment falling bodies law of inertia telescope heliocentric moons sunspots inquisition N

Newton