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Lecture 1: Units, physical quantities
and vectors (Part 1)
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Objectives
• Convert measurements into different units
• Use dimensional analysis in checking the
correctness of an equation
• Differentiate vector and scalar quantities
• Rewrite a vector in component form
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Example:
Units are very important!
Physical quantity
Physical quantity is any number that is used to
describe a physical phenomenon.
Physical quantity = number + unit
(magnitude) (standard)
Time 60 seconds
Length 1.0 meter
Mass 50 kilograms
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Fundamental units
International system (SI or the “metric” system)
Repeatability of measurements
Table 1. SI Base Units
Quantity Name of Unit Symbol
Length meter m
Mass kilogram kg
Time second s
Electric current ampere A
Thermodynamic temperature kelvin K
Amount of substance mole mol
Luminous intensity candela cd 4
Unit prefixes
Easy to introduce larger or smaller units
Multiples of 10 or 1/10
1 nano- = 10-9
1 micro- = 10-6
1 milli- = 10-3
1 centi- = 10-2
1 kilo- = 103
1 mega- = 106 5
Unit consistency and conversion
Equations express relationships among physical
quantities.
Equations must be dimensionally consistent.
Examples:
Height = 163 cm = 1.63 x 102 m = 5 ft, 4 in
Length: 𝑙 = 100 m + 0.25 in
Acceleration: 𝑎 = 𝑣𝑡 = (2m s )(0.26hr) 6
Sample problem: Unit conversion The official world land speed record is 1228.0km/h
set on October 15, 1997 by Andy Green in the jet
engine car Thrust SSC. Express this speed in
meters per second.
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1228.0𝑘𝑚
ℎ= 1288.0
𝑘𝑚
ℎ
1000𝑚
1𝑘𝑚
1ℎ
3600𝑠= 341.1
𝑚
𝑠
Most physical quantities can be expressed in
terms of fundamental dimensions []:
[Length] L
[Time] T
[Mass] M
[Current] A
[Temperature] o
[Amount] N
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DIMENSIONAL ANALYSIS - check if the equation is dimensionally correct
- know the units or the dimension of a physical quantity
Example: Dimensional Analysis
Check whether the following equations is correct:
1. 𝑠 = 𝑣𝑡
2. 𝑣 = 𝑚 + 2𝑎𝑠
Use: 𝑠 = 𝐿𝑒𝑛𝑔𝑡ℎ m = Mass
𝑣 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒 𝑡 = 𝑇𝑖𝑚𝑒
𝑎 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒2
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*substitute dimensions of the physical
quantities
*simplify the dimension of the LHS and
RHS of the equation
*check if the dimension is consistent
Exercise: Dimensional Analysis
Check whether the following equations is correct:
1. 𝑠 = 𝑣𝑡
2. 𝑣 = 𝑚 + 2𝑎𝑠
RHS:
𝑣𝑡 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒𝑇𝑖𝑚𝑒 = Length
LHS:
s = Length
LHS:
𝑣 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒
RHS:
m+ 𝑎𝑠 = 𝑀𝑎𝑠𝑠 +𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒2𝐿𝑒𝑛𝑔𝑡ℎ
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Scalar
A scalar is a quantity that is described by a
number.
Example: 𝑚 = 5 kg, 𝑡 = 60 s
Magnitude of a vector is a scalar (number) and is
always positive.
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Scalar and Vector
Vector
A vector is a quantity that has both magnitude and
direction (displacement, velocity, force).
Example: 𝑥 = 45 to the east, 𝑥 = 45, 26° north of east 12
Vector from the movie Despicable Me, because
he’s “committing crimes with both magnitude and
direction!”
Vector notation
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Describing a vector 1. Bearing
Angle with respect to a chosen axis
4 m
θ = 45° 𝑥
𝑦
Displacement, 𝑑 4 m, 45° north of east 4 m, 45° with respect to the horizontal
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Describing a vector
2. Component form: (𝑥, 𝑦, 𝑧) coordinates
Unit vectors = “1” magnitude
𝒊 , 𝒋 , 𝒌 unit vectors
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Components of a vector
In general form:
𝑷 = Px𝑖 + Py𝑗 + 𝑃𝑧𝑘
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Components of a vector
components of vector P in
x-, y- and z-axis
In general form:
𝑷 = Px𝑖 + Py𝑗 + 𝑃𝑧𝑘
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Components of a vector
unit vectors in
x-, y- and z-axis
components of vector P in
x-, y- and z-axis
In general form:
𝑷 = Px𝑖 + Py𝑗 + 𝑃𝑧𝑘
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Components of a vector
Pz
Px
Py
http://www.intmath.com/vectors/7-vectors-in-3d-space.php
𝑷 = Px𝑖 + Py𝑗 + 𝑃𝑧𝑘
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Px = 2m/s
Py = 3m/s
Pz = 5m/s
𝑷 = 2m/s𝑖 + 3m/𝑠𝑗 + 5𝑚/𝑠𝑘
sometimes
simply
written as
“x”, “y” & “z”
How to calculate the components of a vector
4m
θ = 45° 𝑥
𝑦
x-component
y-component
Components form a right triangle
𝑟𝑥 or 𝑥 = 𝑟𝑐𝑜𝑠𝜃 𝑟𝑦 or 𝑦 = 𝑟𝑠𝑖𝑛𝜃
𝑟 = 𝑥2 + 𝑦2 𝜃 = tan−1𝑦𝑥
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𝒓
Given 𝑟 = 4m, 𝜃 = 45°,
𝑥 = 4m cos 45° = 2 2m
𝑦 = 4m sin 45° = 2 2m 𝒓 = x𝒊 + 𝒚𝒋 + 𝒛𝒌
𝒓 = 𝟐 𝟐m𝒊 + 𝟐 𝟐m𝒋 component form of 𝒓 21
How to calculate the components of a vector
4m
θ = 45° 𝑥
𝑦
x-component
y-component
𝒓
𝒓 = 𝟐 𝟐m𝒊 + 𝟐 𝟐m𝒋 component form of r
(vector form)
What is the magnitude of r?
𝒓 = 𝒙𝟐 + 𝒚𝟐 = (𝟐 𝟐𝒎)𝟐+(𝟐 𝟐𝒎)𝟐 = 𝟒𝒎 22
How to calculate the components of a vector
4m
θ = 45° 𝑥
𝑦
x-component
y-component
𝒓
Example: vectors in bearing
and component form
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Bearing form:
𝑨 = 8.00m, South
𝑩 = 15.0m, 30.0o East
of North
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Example: vectors in bearing
and component form
Component form:
𝑨 = - 8.00m𝒋
for 𝑩:
𝑩𝒙 = 𝟏𝟓. 𝟎𝒎 𝒔𝒊𝒏𝟑𝟎 = 𝟕. 𝟓𝟎𝒎 𝑩𝒚 = 𝟏𝟓. 𝟎𝒎 𝒄𝒐𝒔𝟑𝟎 = 𝟏𝟑. 𝟎𝒎
𝑩 = 7.5m𝒊 + 13.0m𝒋
Bx
By
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Example: vectors in bearing
and component form
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𝒂 = 𝟏. 𝟐𝒎𝒊 + 𝟕. 𝟏𝒎𝒋
Given the components, how to get the
magnitude and direction from a vector?
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𝒂 = 𝟏. 𝟐𝒎𝒊 + 𝟕. 𝟏𝒎𝒋
ax ay
Recall the general form:
𝒂 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘
Given the components, how to get the
magnitude and direction from a vector?
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𝑎 = 𝑎𝑥2 + 𝑎𝑦
2 (manitude of vector a)
𝜃 = tan−1 𝑎𝑦 𝑎𝑥 (angle of vector a)
𝒂 = 𝟏. 𝟐𝒎𝒊 + 𝟕. 𝟏𝒎𝒋
ax ay
Given the components, how to get the
magnitude and direction from a vector?
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𝒂 = 𝟏. 𝟐𝒎𝒊 + 𝟕. 𝟏𝒎𝒋
𝑎 = 𝑎𝑥2 + 𝑎𝑦
2 = (1.2𝑚)2+(7.1𝑚)2= 7.2𝑚
𝜃 = tan−1 𝑎𝑦 𝑎𝑥 = tan−1 7.1𝑚
1.2𝑚 = 80.4o, 260.4°
𝒂 = 7.2m,
80.4o from horizontal
x
y
Given the components, how to get the
magnitude and direction from a vector?
Useful trick:
Any angles that differ by
180O have the same
tangent…
80.4 and 260.4 are tan-1(5.9).
Seatwork
- solve problems in your
notebooks
- write the answers only in
your bluebook
- indicate the date
August 8, 2014
1. Blah?
2. Blah blah!
3. Blah blah blah!
4. Blah blah blah blah!
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1.Determine the dimension on the
LHS and RHS of the equation:
𝑠 = 𝑣𝑡 +1
2𝑎𝑡2
𝑠 = 𝐿𝑒𝑛𝑔𝑡ℎ m = Mass
𝑣 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒 𝑡 = 𝑇𝑖𝑚𝑒
𝑎 =𝐿𝑒𝑛𝑔𝑡ℎ
𝑇𝑖𝑚𝑒2
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Cy
Dx
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2 and 3: Write vector 𝑪 and
𝑫 bearing form:
4 and 5: Write vector 𝑪 and
𝑫 component form:
Dy
Cx
6 and 7. What is the magnitude and
direction of vector q (include which
quadrant)?
𝑞 = −2𝑚𝑖 + 4𝑚𝑗
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Hint: sketch vector q
your CRS email address will be used for
sending the following:
Lecture 1 Slides (to be emailed later)
Problem Set # 1 (to be emailed next week)
(use the last page of your bluebooks
in answering the problems sets,
include solutions)
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