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Sunday, May 30, 2010

Scalars and Vectors: Chapter 3, X Physics,Physics tutors Karachi

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Physical Quantities:
All those quantities which can be measured are called physical quantities. Physical
Quantities can be measured by means of magnitude and units.
Magnitude:
A symbol which gives us the quantity of substance is called magnitude.
Unit:
A symbol which gives its relation is called unit. e.g.
5 KG 5: Magnitude Kg: Unit
10 N 10: Magnitude N : Unit
Type of Physical Quantities:
Physical quantities are of two types.
(i) Scalar quantities (ii) Vector Quantities
(i) Scalar Quantities:
Definition:
“All those physical quantities which are completely specified when only magnitude with
suitable units are given are called scalar quantities”.
OR
“Quantities which can be specified by number having appropriate units are scalar
quantities”.
Example:
Mass, Distance, Time, Speed, Energy, Temperature, Work, Volume, density, etc.
(ii) Vector Quantities:
“All those physical quantities which are completely specified only when both
magnitude with suitable units as well as direction are called vector quantity”.
OR
“Quantities having both magnitude and direction with appropriate units are called
vector quantities”.
Example:
Displacement, velocity, acceleration, force, weight, momentum, torque, etc.
Vector Representation:
A vector is represented graphically by a directed line segment or an arrow head
segment. The length of arrow represents the magnitude while arrowhead represents
the direction. Representation of vector can be performed in three steps.
Step I: “Suitable Scale”
In this step we choose suitable scale in order to represent the given vector.


A
Step II: “ Reference axis”
In this step we represent the direction of given vector in reference axis. N
W E

S

Step III: “Representation”

In this step we represent the given vector by the help of above choosen scale and
indicated direction.
Negative of a vector:
Negative vector is defined as:
“A vector just equal in magnitude but exactly opposite in direction is called negative of
a vector.”
OR
“A vector having the same magnitude as that of a given vector but opposite in
direction is called negative of a vector.”
Example:
Negative of a A
Vector A is –A - A
Addition of Vectors by Head to Tail Rule:
Consider two vector A and B represented by lines OP and OQ respectively as shown in
figure we can and these two vectors by placing the tail of second vector to the head of
first vector. Now join the tail of first vector to the head of last vector, it gives us the
resultant vector as shown in Fig.



B
B P
P
B A
If more than two vectors are to be added than same method will be adopted for that.
First draw the first vector than place the second vector so that its tail is at the head of
first vector. Now add all the vectors just by joining the head of first vector to the tail of
next vector and in last join the tail of first vector to the head of last vector it gives us
resultant vector as shown in figure.
D



C
A
B
Resultant Vector:
“A vector which joins the tail of first vector to the head of last vector is called resultant
vector.”
OR
“ A single vector which gives the combined effect of all the vectors which are to be
added is called resultant vectors.”
It can be represented by the following equation:
R = A + B + C + ………
Where R is the resultant vector, while A, B, C, are vectors to be added.
Subtraction of Vector:
Vectors cannot be subtracted directly. They are subtracted by means of addition. To
subtract vector from another vector sign of the vector is changed and then added to
the other vector. For example if a vector B is to be subtracted from a Vecto: A then A-B
is found by adding vector A and –B. Subtraction of vectors can be illustrated as follows:
A
B -B - B
A



Trigonometry:
Trigonometry is an important branch of mathematics and is used to solve various
problems in Physics. Considering the right angle triangle ABC,
angle and
In this triangle. C
0
A B
Base: (AB) Side adjacent to the angle 0 is called Base.
Perpandicular: (BC) Side opposite to the angle 0 is called Perpendicular.
Hypotaneous: (AC) Largest side or the side opposite to the right angle is called
Hypotaneous
The ratios between any two sides of the right angled triangle is represented by
different names. Some of important ratios are as follows:
Sin 0 = Perp / Hyp Cosec 0 = Hyp / Perp
Cos 0 = Base / Hyp Sec 0 = Hyp / Base
Tan 0 = Perp / Base Cot 0 = Base / Perp
The values of trigonometric ratios are changed if 0 is changed.
Resolution of Vector:
The process of splitting a vector into its parts (components) is called resolution of a
vector. Generally a vector is reloved into two components at right angle to each other
Such components are called rectangular components.
Horizontal Comp: The component which is along horizontal direction is called
Horizontal component.
Vertical Comp: The Component which is along vertical direction is called vertical
Component.
Consider a vector F, which shows the representative line AB making angle 0 with x-axis
from B draw perpendicular BC on x-axis. Suppose AB and AC are represented by two
vectors. Vector BC is parallel to y-axis and y
vector AC is along x-axis. Hence we denote
vector AC by Fx and vector AB by Fy by
applying head to tail rule of vector addition,
the sum of vectors Fx and Fy is equal to F.
Therefore, Fx and Fy are rectangular components B
of Vector F.
F
The magnitude of these components can be Fy
determined by using trigonometric ratios.
Now considering right angle triangle ABC. A x
Fx C
HOR Comp / X Comp: VER Comp / Y Comp:
Cos 0 = Base / Hyp Sin 0 = Perp / Hyp
Cos 0 = AC / AB Cos 0 = BC / AB
But But
AC = Fx1 AB = F BC = Fy1 AB = F
Cos 0 = Fx / F Sin0 = Fy / F
Fx = FCos0 Fy = Fsin0
Addition of Rectangular Components of Vectors:
OR
Composition of Vector:
Rectangular components of vector (components that are perpendicular to each other)
can be join together to form resultant vector or original vector.
Considering right angle triangle ABC. C
Where:
Fx = AB = Base F Fy
Fy = BC = Perp
F = AC = Hyp
For magnitude of vector using pythagorons theorem. 0
(H)2 = (B)2 + (P)2 A B
(AC)2 = (AB)2 + (BC)2 Fx
¾¾¾¾¾¾¾
AC = Ö (AB)2 + (BC)2
OR
¾¾¾¾¾
ÖF = Fx2 + Fy2
For direction of vector using trigonometric ratio:
Tan 0 = Perp / Base
Tan 0 = BC / AC
Tan 0 = Fy / Fx
0 = Tan-1 (Fy / Fx)

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