Transcript Newtonian kinematics - Sierra College Astronomy Home Page

An exposure to
Newtonian mechanics
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Quantities such as velocity, etc.
Newtonian spacetime
Newton’s laws
A smattering of useful equations
Motivation
In 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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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.
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.
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Ex: In an accelerating car.
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)
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.”
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