chapter 1 units and vectors

7/21/2019 Chapter 1 Units and Vectors

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Lecture in Physics 2014-2015

Physics 1

for

Engineer ing StudentsDr. Aladd in Abdul-Lat i f

Physics Department

German University in Cairo

Chapter 1



Units of Chapter 1

•Physical quantities

•Units, Standards, and the SI System

• Converting Units

•Dimensions and Dimensional Analysis

Vectors and ScalarsAddition of Vectors – Graphical Methods

Subtraction of Vectors, and Multiplication of a Vector by

a Scalar

Adding Vectors by Components



1.1 The physical quantities

The laws of physics are expressed in terms of

basic quantities called the physical quantities

Examp les of phys ical quant i t ies

(1) Mass, (2) Length ,

(3) Time, (4)Velocity

(5) Area (6)Volume(7) The density, (8) Acceleration,

(9) Momentum, (10) Force

(11) Work (12) Energy



Types of physical quant i t ies

Fundamental quantities which are definedin terms of measurements or comparison

with established standards.

Derived quantities, which are expressed in

terms of the fundamental quantities.The fundamental quant i t ies in mechanics

In mechanics, the three fundamental

quantities are the length (L), the mass (M),

and the time (T). All other physical

quantities in mechanics can be expressed in

terms of these three quantities.



(a) In the international system of units (SI system)

the length has unit of meter ……..(m),

the mass has unit of kilogram…..(kg),

the time has unit of second ……..( s ).

(b) In the Gaussian system of units (CGS system),

the length has unit of centimeter ……..(cm),

the mass has unit of gram……………...(gm),

the time has unit of second …………….( s ).

(c) In the British system of units (FPS system),the length has unit of foot ……..(ft),

the mass has unit of pound……(p),

the time has unit of second ……(s).

Systems of un i ts



The Standards of Units in the SI System

Quantity Unit Standard

Length Meter Length of the path traveled by

light in 1/299,792,458 second.

Time Second Time required for 9,192,631,770

periods of radiation emitted by

cesium atoms

Mass Kilogram Platinum cylinder in International

Bureau of Weights and Measures,

Paris



We will be working in the SI system, where the

basic units are kilograms, meters, and

seconds.

Other systems: cgs; units are

grams, centimeters, andseconds.

British engineering system has

force instead of mass as one of

its basic quantities, which are

feet, pounds, and seconds.



Convers ion o f uni ts

• To convert units from one system to

another, the conversion factors are as

follows

• 1 mile = 1609 m =1.609 km

• 1 ft = 0.3048 m = 30.48 cm

• 1 m = 39.37 in = 3.28 ft

• 1 in = 2.54 cm



Unit prefixes SI System

In the metric (SI) system, thelarger and smaller units are

defined in multiples of 10

from the standard unit.

In the given table we give the

standard SI prefixes for

indicating powers of 10. Many

are familiar; Y, Z, E, h, da, a, z,and y are rarely used.



Dimensional analysis

• The word dimension is used in physics to denote

the nature of a physical quantity regardless ofthe system of units used.

• For example whether a distance is measured inunits of feet or meters, it is a distance. We say

its dimension is length (L).• We use square brackets [ ] to denote the

dimension of a physical quantity.

• For example

[mass]=[m]=M, [length] = [l] = L and [time]=[t ]=T.

• Another example is [velocity] =LT-1



The method of d imens ional analys is

This method is used to(a) derive the dimensions of a physical quantity

e.g. [force]=[m a]= MLT-2,

(b) derivation of physical formulas and

expressions.(c) check the correctness of formula or expression.

In (b) and ( c ) we use the fact that in equations

and formulas the left hand side (LHS) must havethe same dimensions as the right hand side(RHS)



Examples|||Example 1.|||

Use the method of dimensional analysis method to show that x=(1/2)at2 is dimensionallycorrect where x is the displacement, a acceleration, and t is the time

Solution : The dimension of LHS [x]=L,

The dimension of RHS [at2]=LT-2.T2 =L

|||Example 2|||

Deduce the equation for the velocity of sound v in a gas, at constant temperature, given

that it depends on the pressure P and the density r of the gas.Solution:

Let v = (const) x Pa x r b

[v] = LT -1 =[P a r b ] = [ML-1T -2 ] a [ML-3 ] b

LT -1 = M a+b L-a-3b T -2a

a+b=0, -a-3b=-1, and 2a= -1

a= -1/2 , b=1/2, i.e. v=(const) x P-1/2 x r 1/2



Vectors and Scalars

A vector has magnitude as

well as direction.

Some vector quantities:

displacement, velocity, force,

momentumA scalar has only a magnitude.

Some scalar quantities: mass,

time, temperature



Addition of Vectors – Graphical Methods

For vectors in one

dimension, simple

addition and subtraction

are all that is needed. You do need to be careful

about the signs, as the

figure indicates.




If the motion is in two dimensions, the situation is

somewhat more complicated.

Here, the actual travel paths are at right angles to

one another; we can find the displacement by

using the Pythagorean Theorem.




Adding the vectors in the opposite order gives the

same result:




Even if the vectors are not at right

angles, they can be added graphically byusing the “Head - to tail method”




The parallelogram method may also be used;

here again the vectors must be “tail-to-tip.”



Subtraction of Vectors, and

Multiplication of a Vector by a Scalar

In order to subtract vectors, wedefine the negative of a vector, which

has the same magnitude but points

in the opposite direction.

Then we add the negative vector:



Multiplication of a Vector by a Scalar

A vector V can be multiplied by a scalar m; the

result is a vector mV that has the same direction

as V but a magnitude mV . If m is negative, the

resultant vector points in the opposite direction.

Addi V t b C t




Any vector can be expressed as the sum

of two other vectors, which are called its

components. Usually the other vectors are

chosen so that they are perpendicular to

each other.

Addi V t b C t




If the components are

perpendicular , they can be found

using trigonometric functions.

Addi V t b C t




The components are effectively one-dimensional,

so they can be added arithmetically:

Addi V t b C t




Adding vectors:

1. Draw a diagram; add the vectors graphically.

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using sines and cosines.

5. Add the components in each direction.

6. To find the length and direction of the vector, use:



© Aladdin A bdul-Lat i f 25

Unit Vector

It is a vector having unit magnitude . If A is vector with

magnitude A = 0 then a = A / A is a unit vector having thesame direction as A. The vector A can be represented by

A= A a.

The Rectangular vecto r i , j and k

In the rectangular coordinate system we have the followingunit vectors:

the unit vector i is in the direction of positive x-axis ,

the unit vector j is in the direction of positive y-axis and

the unit vector k is in the direction of positive z-axis .

In the x-y plane we have only i and j as unit vectors.




Resolution of Vector

Any vector A in x-y plane can be resolved into two

components perpendicular to each others one in x-direction and one in y-direction, See fig 7.

The x-component Ax is given by:

A x = A co s

The y-component Ay is given by:

A y = A s in

where q is the angle between the vector A and thepositive x- axis

y

x

A




Vector Equation




Sum of two Vectors

• Addition of two vectors by the analytical method:

Let A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k, thenC = A + B = ( Ax + Bx )i + ( Ay + By ) j + ( Az + Bz ) k

Since C can be written as C = Cx i + Cy j + Cz k , then

C x = ( Ax + Bx ),

C y = ( Ay + By ), and

C z =( Az + Bz )In x-y plane we have only two components Cx and Cy thatgives for the magnitude of C the value

C 2 =C x 2 + C y

2




Dot or Scalar Product




Cross or Vector Product

The cross or vector product of A and B is a vector

C = A x B( read A cross B ).The magnitude of A x Bis defined as the product of the magnitude of A and

B and the sine of the angle q between them. The

direction of the vector C = A x B is perpendicular to

the plane of A and B such that A , B and C form a

right - handed system . In symbols

A x B = AB sin q ç

where ç is a unit vector indicating the direction ofA x B. If A = B , or if A is parallel to B , then sin q = and

we define A x B = 0




A second kind of vector multiplication is the vector product (also called the

cross product), defined as

A x В = С

where the magnitude of С is ICI = AB sin q

Cross Product

q, the angle between A and B, lies between 0° and 180°. С is

directed perpendicular to the plane of A and B.

To determine the direction of C, use the following right-handrule (see the Given Figure).

(a) Slide A and В to-together so their tails touch. Next,

(b) using your right hand, point your fingers along A.

(c) Curl your fingers toward B.

(d) Your right thumb will point up in the direction of

С = A x В. (e) С points in the direction a right-hand screw would

advance when A is rotated toward B. Note that

В x A = - A x B.If A and В are parallel, q = 0 and | A x B| = 0. If A and В are

perpendicular, q = 90°, sin 90° = 1, and |A x B| = AB.




Cross or Vector Product



7- The magnitude of =I I= IA x BI is the same as the

area of a parallelogram with sides A and B

8- If A x B = 0 , and A and B are not null vectors , then A

and B are parallel .

chapter 1 units and vectors

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