Introduction and Mathematical Concepts Chapter 1.
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Transcript of Introduction and Mathematical Concepts Chapter 1.
![Page 1: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/1.jpg)
Introduction and Mathematical Concepts
Chapter 1
![Page 2: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/2.jpg)
1.1 The Nature of Physics
Physics has developed out of the effortsof men and women to explain our physicalenvironment.
Physics encompasses a remarkable variety of phenomena:
planetary orbitsradio and TV wavesmagnetismlasers
many more!
![Page 3: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/3.jpg)
1.2 Units
Physics experiments involve the measurementof a variety of quantities.
These measurements should be accurate andreproducible.
The first step in ensuring accuracy andreproducibility is defining the units in whichthe measurements are made.
![Page 4: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/4.jpg)
1.2 Units
SI units
meter (m): unit of length
kilogram (kg): unit of mass
second (s): unit of time
![Page 5: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/5.jpg)
1.2 Units
![Page 6: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/6.jpg)
1.2 Units
The units for length, mass, and time (aswell as a few others), are regarded asbase SI units.
These units are used in combination to define additional units for other important
physical quantities such as force and energy.
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1.5 Scalars and Vectors
A scalar quantity is one that can be describedby a single number:
temperature, speed, mass
A vector quantity deals inherently with both magnitude and direction:
velocity, force, displacement
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1.6 Vector Addition and Subtraction
Often it is necessary to add one vector to another.
![Page 9: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/9.jpg)
1.6 Vector Addition and Subtraction
![Page 10: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/10.jpg)
1.6 Vector Addition and Subtraction
When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.
![Page 11: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/11.jpg)
1.7 The Components of a Vector
. ofcomponent vector theand
component vector thecalled are and
r
yx
y
x
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1.7 The Components of a Vector
.AAA
AA
A
yx
that soy vectoriall together add and
axes, and the toparallel are that and vectors
larperpendicu twoare of components vector The
yxyx
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1.7 The Components of a Vector
It is often easier to work with the scalar components rather than the vector components.
. of
componentsscalar theare and
A
yx AA
1. magnitude with rsunit vecto are ˆ and ˆ yx
yxA ˆˆ yx AA
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1.8 Addition of Vectors by Means of Components
BAC
yxA ˆˆ yx AA
yxB ˆˆ yx BB
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1.8 Addition of Vectors by Means of Components
yx
yxyxC
ˆˆ
ˆˆˆˆ
yyxx
yxyx
BABA
BBAA
xxx BAC yyy BAC
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Example 1.1
• A vector A has a magnitude of 20cm at 200 , and a vector ⁰ B has a magnitude of 37cm at 45⁰. What is the magnitude and direction of vector difference A – B?
• Solution:
Vector Distance Angle X-component Y-component
A 20cm 200⁰ Ax = 20cos200⁰ Ay = 20sin200⁰
B 37cm 45⁰ Bx = 37cos45⁰ By = 37sin45⁰
A – B -44.95cm -33.00cm
![Page 17: Introduction and Mathematical Concepts Chapter 1.](https://reader030.fdocuments.us/reader030/viewer/2022033013/56649d355503460f94a0c304/html5/thumbnails/17.jpg)
• Magnitude: |A - B|= [(Ax – Bx)2 + (Ay – By)2]
= [(-44.95)2 + (-33.00)2]2 = 55.76cm
• Direction: θ = tan-1 [(Ay – By)]/[(Ax – Bx)]
= [-33.00/-44.95] = 36.28⁰ = 216.28⁰
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Problems to be solved
• 1.36, 1.51, 1.52, 1.57, 1.63, 1.65
• B1.1: A disoriented physics professor drives 4.92km east, then 3.95km south, then 1.80km west. Find the magnitude and direction of the resultant displacement, using the method of components.
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• B1.2: The three finalists in a contest are brought to the centre of a large, flat field. Each is given a meter stick, a compass, a calculator, a shovel, and (in a different order for each) the following three displacements: 72.4m, 32⁰ east of north, 57.3m, 36⁰ south of west, 17.8m straight south. The three displacements lead to the point where the keys to a new Porsche are buried. Two contestants start measuring immediately, but the winner first calculates where to go. What does he calculate?
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• B1.3: After an airplane takes off, it travels 10.4km west, 8.7km north, and 2.1km up. How far is it from the take-off point?