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An exposure to Newtonian mechanics• Quantities such as velocity, etc.

• Newtonian spacetime• Newton’s laws• A smattering of useful equations

MotivationIn order to understand aspects of more modern physics and astronomy, we must ground it in the context of conventional science.

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The 3+1 Newtonian grid of the Universe

Space consists of three spatial dimensions.

To every point in space, you can assign coordinates (x, y, z).

This is your reference frame.

Distances between points in this space are given by the Pythagorean theorem,

L2=Δx2 + Δy2 + Δz2

Absolute space, in its own nature, without regard to anything external, remains always similar and immovable.

– Newton (1687)

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The 3+1 Newtonian grid of the Universe

Time (t) is the beat to which the Universe ages.

One can assign the same time value to all points in the grid, thus establishing when things are simultaneous, and when they are not.

The rate of time is 1 sec/sec, everywhere.

Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration....

– Newton (1687)

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Linear Velocity and Acceleration

If an object changes (Δ) the value of its spatial coordinate, during a certain interval of time, we call this velocity.

vx= Δx/Δt vy= Δy/Δt vz= Δz/Δt

Note: velocity is not quite the same as speed. It includes direction information: it is a vector, not a scalar.

If an object changes the value of its velocity during a certain interval of time, we call this acceleration.

ax= Δvx/Δt ay=Δvy/Δt az=Δvz/Δt

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Linear Velocity and Acceleration

Close your eyes in a luxury car.

You cannot feel velocity. You can feel acceleration.

The physical laws of the world work if whether we are sitting still, or moving in a straight line.

—Or, stated another way…

Physical laws work the same whether your reference frame is…

— stationary to someone else’s

— moving with respect to someone else’s.

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Linear Velocity and Acceleration

Velocities add together quite sensibly.

vtotal= v1 + v2

This is called Velocity Addition.

Many other vectors and scalars add this way too, such as accelerations, changes in distance, etc.

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This law defines a reference frame that is either stationary with respect to the observer (v=0), or moving (v=0).

Newton’s laws work within an inertial reference frame. If an object changes its velocity WITHOUT an external force acting on it, you are not in an inertial reference frame.

Ex: In an accelerating car.

Newton’s Laws of Motion

LAW #1: An object at rest stays at rest, unless acted upon by an external, unbalanced force.

Similarly, an object in motion continues in motion, unless acted upon by an external, unbalanced force.

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Inertial reference frames

A belief in absolute reference frames is so ingrained into our minds, they have become a metaphor for comforting reliability.

I have of late,—but wherefore I know not,—lost all my mirth, forgone all custom of exercises; and indeed, it goes so heavily with my disposition that this goodly frame, the earth, seems to me a sterile promontory; this most excellent canopy, the air, look you, this brave o’erhanging firmament, this majestical roof fretted with golden fire,—why, it appears no other thing to me than a foul and pestilent congregation of vapours.

—Shakespeare, “Hamlet,” Act II, scene ii

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Mass, Momentum, and Force

Mass is a measure of how much stuff you are dealing with. It is (in Newton’s framework) a sum of the particles that constitute something. It is NOT weight.

M, m, etc. is measured in kg.

Momentum is a useful concept, p=mv.

Force is a measure of how hard one thing pushes or pulls upon something else.

F is measured in Newtons or pounds.

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Newton’s Laws of Motion

LAW #2: If an unbalanced force (F) acts upon a mass (m), the mass will respond with an acceleration (a), such that:

F = ma

This law can be rewritten:

Since a= Δv/Δt

F = ma = m(Δv/Δt)

= (mΔv)/Δt

= Δp/Δt

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Newton’s Laws of Motion

LAW #3: For every force, there is a counter force of equal strength but opposite direction:

F1 = -F2

m1a1 = -m2a2

This law can be manipulated to:

a1 = -(m2/m1)a2

This does not mean that nothing can happen!

Ex: Rockets, bullets, or energy dissipation.

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Newton’s Laws of Motion

Law of Gravity: For two objects, of masses m1 and m2, with their centers separated by distance R12, there is a mutually attractive

force :

F12 = Gm1m2/R122

Notes

G=6.67×10-11 N m2/kg2

Gravity never goes away.

Gravity is an inverse-square law.

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Energy in various forms

Kinetic energy (K.E.)

1) K.E.=½mv2

2) Temperature is a measure of KE/particle

Potential energy (P.E.)

3) P.E.=mgh

(falling under the influence of a uniform gravitational field)

4) P.E.=-GMm/R

(approaching a massive object from very far away)

More forms of P.E. exist: batteries, springs, etc.

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Circular Motion

Linear motion

x, v, a, F, p, E, t

Angular motion

θ, ω, α, τ, L, E, t

Force required to keep an object in circular motion:

Fc = mvc2/R

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Conservation Laws

A vast number of problems in physics are solved using principles of conservation.

Linear momentum and total energy are conserved.

Notes

Angular momentum is also a conserved quantity.

Energy is conserved, but it may be transferred to/from K.E., P.E., and other things.

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Things Worth Noting!Newton’s genius is reflected in three insightful assumptions.

1)Action at a distance for the Law of Gravity.

2)Inertial mass (F=ma) is the same thing that causes gravity (F=Gm1m2/R2).

3)Previous to Newton, scientists treated terrestrial physics and cosmic physics differently. Newton uses the same laws for both settings. Showing that both sets of laws are really aspects of a simpler, underlying single set of laws, is called “Unification.”