Introduction and Mathematical Concepts

Chapter 1

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!

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.

1.2 Units

SI units

meter (m): unit of length

kilogram (kg): unit of mass

second (s): unit of time

1.2 Units

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.

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

1.6 Vector Addition and Subtraction

Often it is necessary to add one vector to another.

1.6 Vector Addition and Subtraction

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.

1.7 The Components of a Vector

. ofcomponent vector theand

component vector thecalled are and

r

yx

y

x

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

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

1.8 Addition of Vectors by Means of Components

BAC

yxA ˆˆ yx AA

yxB ˆˆ yx BB

1.8 Addition of Vectors by Means of Components

yx

yxyxC

ˆˆ

ˆˆˆˆ

yyxx

yxyx

BABA

BBAA

xxx BAC yyy BAC

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

• 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

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.

• 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?

• 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?