1stle lecture 01 - r1 units, physical quantities and vectors part 1a.pdf

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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