Notes, solutions, and commentary on Physics For Mathematicians Mechanics I by Micheal Spivak.
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1.1 Basic Concepts
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1.1.1 Mass and Force
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Newton does not offer operational definitions of mass or force, as such they are left as vague intuitive concepts.
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Mass : :
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In the Principia Newton calls mass a 'quantity of matter'.
Quantity of matter is a measure of matter that arises from its density and volume jointly.
'Quantity of matter' invokes the notion of 'physical quantity' native to contemporary physics. A physical quantity is a numerical value with dimensions, it is meant to model some aspect of a physical system.
In this case, density and volume are the dimensional aspects of Newton's 'quantity of matter. Using volume and density makes some sense given one would think that the quantity of matter, whatever it is, would be related to the amount of stuff, that is density, in a unit of space, in this case a volume (3d space).
Under the lens of modern physics such a definition is viewed as unsatisfactory because it is circular given density is defined as the ratio of mass to volume.
A modern definition of mass is what newton called "inertia of mass".
Inertia of mass is a body's resistance to being moved if it is at rest - or of having its velocity changed if it is already moving with uniform velocity.
The inertia of mass definition is less circular. If we had two spheres, one made of cork and the other of iron, that are the same size, we would have to exert a great deal more force on the iron ball than the cork ball to get it moving.
Hence, the inertia of mass is reliant on the notion of force and resistance and is defined as the resistance to being moved by a force.
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Force : :
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Despite 'Force' being one of the most fundamental concepts in all of physics it is also one of the hardest to quantify accurately.
As a mathematical object a force is a R³ vector because it has a magnitude and direction represented by real valued quantities in 3 dimensions.
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Momentum : :
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What Newton called 'Quantity of Motion' we now call momentum of an object.
Momentum is the product of mass, m, and a velocity vector, v.
This definition assumes that the motion of an object can be described as a curve c:R³->R³, with a velocity vector v = c'. This fails to describe something like a ball rolling down an inclined plane or a rod revolving as it is thrown.
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Point Masses : :
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Particles or point masses are abstractions that don't actually exist but allow us to represent bodies in physical systems in an approximate enough way that we can call the results and models 'reasonable'.
As mathematical objects a particle is a path c:R³->R³, with derivative c'=v and a number m > 0 ∈ R.
last edit on
5/29/2022 1:28:07 AM
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Thrall to the Wire of Self-Excited Circuit.
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This is fine. You’d prefer Blanc or JitK or CS spam? You might actually get smarter, not dumber with this.
I'm more of a computer/psych guy. I'll learn mathematic concepts if I need to, but it's not something I actively am interested in. To each their own, I'll at least try to interpret some of this xd
visceral normality
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This is fine. You’d prefer Blanc or JitK or CS spam? You might actually get smarter, not dumber with this.
I'm more of a computer/psych guy. I'll learn mathematic concepts if I need to, but it's not something I actively am interested in. To each their own, I'll at least try to interpret some of this xd
I too am into computer stuff. If you were to find any thread of mine interesting I would pay attention to 'Concrete Mathematics. GKP'. Concrete Mathematics was written as a prerequisite for all the mathematics one would find in Donald Knuths The Art of Computer Programming which is considered by most to be the bible of algorithmic analysis.
1.2 Newtons Three Laws
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1.2.1 The First Law
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Traditional First Law : :
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In the Principia Newton states the first law as,
Every body perseveres 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.
A more contemporary statement,
An object not acted upon by any force has a constant velocity v, and, in particular, if it is initially at rest, the it remains at rest.
Newton states this principal as something "accepted by mathematicians and confirmed by experiments of many kinds", mentioning Galileo especially.
Galileo's contribution to the principal was a compilation of experiments and reasoning for its validity despite seeming contradictory to every day experience at the time.
The law in its contemporary formulation is still incomplete given the notion of velocity and position requires a coordinate system used by an observer.
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A More Complete Definition : :
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Fundamentally, the first law establishes the distinction between spatial coordinate systems.
The first law stats that there are certain coordinate systems for which v is constant unless acted upon by forces.
A coordinate system of this kind is known as a Inertial System as it models a bodies inertia or tendency to have uniform motion unless acted upon by a force.
More complete Definition,
There is at least one coordinate system - an inertial system - in which any object acted upon by any force has constant velocity.
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Philosophical Problems : :
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Any coordinate system moving with a uniform velocity with respect to an inertial system is itself an inertial system.
Things such as the room or space in which an experiment is being completed in are treated as an inertial system but they really are not given that space is rotating over a 24 hour period around an axis, and rotating around the sun over an annual period.
It is hard to establish an actual inertial system.
In Newtons time 'fixed stars' really were thought of as fixed, therefore an actual inertial system relative to those fixed stars was a reasonable view to hold.
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1.2.2 The Second Law
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Second Law : :
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Newton accredits Galileo for the second law as well,
In an inertial system, the rate of change of momentum of a particle is directly proportional to the force F acting on it, F = (mv)' = m.v'
In Principia Newton speaks of the change of momentum instead of its derivative, so the second law is stated in terms of "impulsive forces" that act "instantaneously".
Newton does use the second law in a general sense depending on his needs,
A change in motion [momentum] is proportional to the motive force impressed and takes place a long the straight line in which that force is impressed.
Note that the first law cannot be stated as a special case of F = 0 of the second law because it is the first law that defines an inertial system.
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Second Law as Mathematical Object : :
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F = m.v'
The quantity F that is vague and difficult to measure and as discussed in the last section mass too can be ambiguous.
The equation of the second law mathematically speaking would work better as an axiom.
The reason an ambiguous equation is treated as a physical law is because Newton draws upon the observational and experimental work of Galileo.
"When a body falls, uniform gravity, by acting equally in individual equal particles of time, impresses equal forces upon that body and generates equal velocities; and in total time it impresses a total force and generates a total velocity proportional to the time. And spaces described in proportional times are as the velocities and the times jointly, that is, in the squared ratio of the times."
The passage was inspired by Galileo's observation that the decent of heavy bodies is in the squared ratio of the time.
Newton means we can split time up into intervals Δt = t/N for a large number N and regard the motion with constant acceleration a as being uniform on each interval, with an instantaneous change of speed of a.Δt at the beginning of each interval.
A body starting at rest has velocity aΔt during the first interval Δt falling a distance a(Δt)².
It has the velocity 2aΔt during the next interval t falling a distance 2a(Δt)².
At the end of time t it has fallen a distance of (1+2+3+...+N).a(Δt)² = (1/2)aN(N+1)(Δt)²
Given NΔt = t our answer is ~ (1/2)at².
The modern version,
If s'' = α for a constant α, then s' = αt, thus s = (1/2)αt²
The explanation becomes meaningful given uniform gravity.
Uniform gravity asserts that gravity is a constant and does not vary with distance.
Once the force of gravity is known to be constant F = mc" asserts that the downward acceleration constant which and is verified by the observation that distance traveled by a body at time t is proportional to t².
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