Download Dimensions in Mechanics Base Quantities SI Base Units

Dimensions in Mechanics
Physical quantities have dimensions.
These quantities are the basic dimensions:
- mass, length, time with dimension symbols M, L, T
Other quantities’ dimensions are more complex:
Dimensions, Units, and
Problem Solving Strategies
- [velocity] = length/time = LT-1
- [force] = (mass)(length) /(time)2 = MLT-2
- [any mechanical quantity] = Mα Lβ Tγ
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- where α β, and γ can be negative and/or non-integer
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Base Quantities
Name
Length
Symbol for
quantity
l
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SI Base Units:
Symbol for
dimension
L
SI base
unit
meter
Time
t
T
second
Mass
m
M
kilogram
Electric current
I
I
ampere
Thermodynamic Temperature
T
Θ
kelvin
Amount of substance
n
N
mole
Luminous intensity
IV
J
candela
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Second: The second is the duration of
9,192,631,770 periods of the radiation
corresponding to the transition between the
two hyperfine levels of the ground state of the
cesium 133 atom.
Meter: The meter (m) is now defined as the
distance traveled by a light wave in vacuum
in 1/299,792,458 seconds.
Mass: The SI standard of mass is a
platinum-iridium cylinder assigned a mass
of one 1 kg
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Speed of Light
In 1983 the General Conference on Weights and
Measures defined the speed of light to be the best
measured value at that time:
c = 299, 792, 458 meters/second
This had the effect that length became a derived
quantity, but the meter was kept around for practicality
Fundamental and Derived
Quantities: Dimensions and Units
The dimensions of (new) physical quantities
follow from the equations that involve them
F = ma
implies that
[Force] = M1 L1 T −2 = M L T −2
However, it is impossible to measure the speed of light
these days!
Since we use force so often, we define new units to
measure it: Newtons, Pounds, Dynes, Troy Oz.
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Table Problem: Dimensions
Solution
Table Problem: Dimensions
Determine the dimensions of the following mechanical
quantitites:
1.
[momentum]
2.
[pressure]
3.
[kinetic energy]
4.
[work]
5.
[power]
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Determine the dimensions of the following mechanical
quantitites:
[momentum] = (mass)(velocity) = M L T -1
[pressure] = [force/area]=M L T-2 / L2 = M L-1 T -2
[kinetic energy] = [(mass)(velocity) 2 ] = M( L T -1 ) 2 = M L2 T -2
[work] = [(force)(distance)] = (M L T -2 )(L) = M L2 T -2
[power] = [(work)/(time)]=M L2 T -2 / T =M L2 T -3
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Dimensional Analysis:
Strategy
When trying to find a dimensional correct formula for a quantity from a set of
given quantities, an answer that is dimensionally correct will scale properly
and is generally off by a constant of order unity
Since:
[desired quantity] = Mα Lβ Tγ where α β, and γ are known
Combine the given quantities correctly so that:
Table Problem: Dimensional
Analysis
The speed of a sail-boat or other craft that does not
plane is limited by the wave it makes – it can’t climb
uphill over the front of the wave. What is the maximum
speed you’d expect?
Hint: relevant quantities might be the length l of the
boat, the density ρ of the water, and the gravitational
acceleration g.
[desired quantity] = Mα Lβ Tγ = (given1)X (given2)Y (given3)Z
vboat = l X ρ Y
- solve for X, Y, Z to match correct dimensions of desired quantity
g
Z
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Dimensions of quantities that
may describe the maximum
speed for boat
Name of Quantity
Symbol
Dimension
Maximum speed
v
LT-1
density
ρ
ML-3
Gravitational
acceleration
g
LT-2
Length
l
L
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Problem Solving
Measure understanding in technical and scientific
courses
Problem solving requires factual and procedural
knowledge, knowledge of numerous schema, plus skill
in overall problem solving.
Schema is loosely defined as a “specific type of problem”
such as principal, rate, and interest problems, onedimensional kinematic problems with constant
acceleration, etc..
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Understand: Get it in Your
Head
Four Stages of Attack
What concepts are involved?
1. Understand the Problem and Models
Represent problem - Draw pictures, graphs, storylines…
2. Plan your Approach – Models and
Similarity to previous problems?
Schema
What known models/physical principles are involved?
3. Execute your plan (does it work?)
e.g. motion with constant acceleartion; two bodies
4. Review - does answer make sense?
Find special features, constraints
e.g. different acceleration before and after some instant
- return to plan if necessary
in time
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Plan your Approach
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Execute the Plan
From general model to specific equations
Model: Real life contains great complexity, so in
physics you actually solve a model problem that
contains the essential elements of the real problem.
Examine equations of the models, constraints
Is all/enough understanding embodied in Eqs.?
count equations and unknowns
Build on familiar models
simplest way through the algebra
Lessons from previous similar problems
Solve analytically (numbers later)
Select your system,
Keep notation simple (substitute later)
Pick coordinates to your advantage
Keep track of where you are in your plan
Are there constraints, given conditions?
Check that intermediate results make sense
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Stuck?
Review
Represent the Problem in New Way
a. Does the solution make sense?
– Graphical
– Pictures with descriptions
– Pure verbal
– Equations
Check units, special cases, scaling. Is your
answer reasonably close to a simple
estimation?
b. If it seems wrong, review the whole process.
Could You Solve it if…
– the problem were simplified?
– you knew some other fact/relationship?
– You could solve any part of problem, even a
simple one?
c. If it seems right,
review the pattern and models used,
note the approximations,
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Estimation Problems
tricky/helpful math steps.
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Estimations Strategies
Identify a set of quantities that can be estimated or
calculated.
Quantitative estimation of phenomena
Order of magnitude estimations
What type of quantity is being estimated?
How is that quantity related to other quantities, which can
be estimated more accurately?
Establish an approximate or exact relationship between
these quantities and the quantity to be estimated in
the problem
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Concept Question: Estimation
Volume of Earth’s Atmosphere
What is your best estimate for the volume of the earth’s lower
atmosphere that contains two thirds of the air molecules?
1) between 101 and 105 cubic meters
3) between
and
1015
5) between
and
1025
Volume of atmosphere: Vol = 4π Re 2t (4π )(6 × 106 m) 2 (1× 104 m)=((4 ×1018 m3 )
Number of molecules = (number of molecules in a mole)(number of moles in
atmosphere)
Number of molecules in a mole NA = 6 x 1023 molecules/mole.
at STP (Standard Temperature and Pressure): one mole of an ideal gas
occupies 22.4 L = 22.4 x 10-3 m3.
cubic meters
4) between 1015 and 1020 cubic meters
1020
of atmosphere. Estimate thickness of atmosphere as 10 km.
Number of moles in atmosphere = (volume of atmosphere)/(volume of mole)
2) between 105 and 1010 cubic meters
1010
Number of Molecules in
Atmosphere
Estimation quantity is a scalar: N number of molecules and is related to volume
Number of molecules in atmosphere:
N = (6 × 1023 molecules/mole)(4 × 1018 m 3 ) / (22.4 × 10−3 m 3 / mole)
cubic meters
= 1 × 1044 molecules
6) between 1025 and 1030 cubic meters
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Molecules in Atmosphere:
Pressure
Estimation quantity is a scalar: N number of molecules and is
related to the mass of the atmosphere. Determine mass per square
meter from atmospheric pressure. Atmospheric pressure at the surface
of the earth is weight of a column of air per area. Patm = 1 x 105 N m-2.
mass / area ( Pressure / g ) = (1×105 N m-2 ) /(10 m s-2 ) = 104 kg m-2
Total mass equals surface area of earth times mass per square meter.
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mass = (mass / area)(4π Rearth
) (104 kg m-2 )(4π )(6 × 106 m)2 = (4 × 1018 kg)
Number of moles in atmosphere = (mass of atmosphere)/(mass per
mole)
-3
-1
mass per mole (30 ×10 kg mole )
# moles (4 ×1018 kg)/(30 ×10-3 kg mole-1 ) 1.3 ×1020 mole
Number of molecules = (number of molecules in a mole)(number
of moles in atmosphere). Number of molecules in a mole NA = 6 x
1023 molecules/mole.
N = (6 × 1023 molecules/mole)(1.3 × 1020 mole) = 1× 1044 molecules
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