Physics Notes for karachi board..1st Year - FULL NOTES
Vectors::::
SCALARS AND VECTORS.
Scalars and Vectors
Scalars
Physical quantities which can be completely specified by
1. A number which represents the magnitude of the quantity.
2. An appropriate unit are called Scalars.
Scalars quantities can be added, subtracted multiplied and divided by
usual algebraic laws. Examples
Mass, distance, volume, density, time, speed, temperature, energy,
work, potential, entropy, charge etc.
Vectors
Physical quantities which can be completely specified by
1. A number which represents the magnitude of the quantity.
2. An specific direction
are called Vectors.
Special laws are employed for their mutual operation.
Examples
Displacement, force, velocity, acceleration, momentum.
Representation of a Vector
A straight line parallel to the direction of the given vector used to
represent it. Length of the line on a certain scale specifies the
magnitude of the vector. An arrow head is put at one end of the line
to indicate the direction of the given vector.
The tail end O is regarded as initial point of vector R and the head P
is regarded as the terminal point of the vector R.
Unit Vector
A vector whose magnitude is unity (1) and directed along the
direction of a given vector, is called the unit vector of the given
vector.
A unit vector is usually denoted by a letter with a cap over it. For
example if r is the given vector, then r will be the unit vector in
the direction of r such that
r = r .r
Or
r = r / r
unit vector = vector / magnitude of the vector
Equal Vectors
Two vectors having same directions, magnitude and unit are called
equal vectors.
Zero or Null Vector
A vector having zero magnitude and whose initial and terminal points
are same is called a null vector. It is usually denoted by O. The
difference of two equal vectors (same vector) is represented by a
null vector.
R - R - O
Free Vector
A vector which can be displaced parallel to itself and applied at any
point, is known as free vector. It can be specified by giving its
magnitude and any two of the angles between the vector and the
coordinate axes. In 3-D, it is determined by its three projections
on x, y, z-axes.
Position Vector
A vector drawn from the origin to a distinct point in space is called
position vector, since it determines the position of a point P
relative to a fixed point O (origin). It is usually denoted by r. If
xi, yi, zk be the x, y, z components of the position vector r, then
r = xi + yj + zk
Negative of a Vector
The vector A. is called the negative of the vector A, if it has same
magnitude but opposite direction as that of A. The angle between a
vector and its negative vector is always of 180º.
Multiplication of a Vector by a Number
When a vector is multiplied by a positive number the magnitude of the
vector is multiplied by that number. However, direction of the
vector remain same. When a vector is multiplied by a negative
number, the magnitude of the vector is multiplied by that number.
However, direction of a vector becomes opposite. If a vector is
multiplied by zero, the result will be a null vector.
The multiplication of a vector A by two number (m, n) is governed by
the following rules.
1. m A = A m
2. m (n A) = (mn) A
3. (m + n) A = mA + nA
4. m(A + B) = mA + mB
Division of a Vector by a Number (Non-Zero)
If a vector A is divided by a number n, then it means it is
multiplied by the reciprocal of that number i.e. 1/n. The new vector
which is obtained by this division has a magnitude 1/n times of A.
The direction will be same if n is positive and the direction will
be opposite if n is negative.
Resolution of a Vector Into Rectangular Components
Definition
Splitting up a single vector into its rectangular components is
called the Resolution of a vector.
Rectangular Components
Components of a vector making an angle of 90º with each other are
called rectangular components.
Procedure
Let us consider a vector F represented by OA, making an angle O with
the horizontal direction.
Draw perpendicular AB and AC from point on X and Y axes respectively.
Vectors OB and OC represented by Fx and Fy are known as the
rectangular components of F. From head to tail rule of vector
addition.
OA = OB + BA
F = Fx + Fy
To find the magnitude of Fx and Fy, consider the right angled triangle OBA.
Fx / F = Cos ? => Fx = F cos ?
Fy / F = sin ? => Fy = F sin ?
Addition of Vectors by Rectangular Components
Consider two vectors A1 and A2 making angles ?1 and ?2 with x-axis
respectively as shown in figure. A1 and A2 are added by using head to
tail rule to give the resultant vector A.
The addition of two vectors A1 and A2 mentioned in the above figure,
consists of following four steps.
Step 1
For the x-components of A, we add the x-components of A1 and A2 which
are A1x and A2x. If the x-components of A is denoted by Ax then
Ax = A1x + A2x
Taking magnitudes only
Ax = A1x + A2x
Or
Ax = A1 cos ?1 + A2 cos ?2 ................. (1)
Step 2
For the y-components of A, we add the y-components of A1 and A2 which
are A1y and A2y. If the y-components of A is denoted by Ay then
Ay = A1y + A2y
Taking magnitudes only
Ay = A1y + A2y
Or
Ay = A1 sin ?1 + A2 sin ?2 ................. (2)
Step 3
Substituting the value of Ax and Ay from equations (1) and (2)
respectively in equation (3) below, we get the magnitude of the
resultant A
A = |A| = v (Ax)2 + (Ay)2 .................. (3)
Step 4
By applying the trigonometric ratio of tangent ? on triangle OAB, we
can find the direction of the resultant vector A i.e. angle ? which A
makes with the positive x-axis.
tan ? = Ay / Ax
? = tan-1 [Ay / Ax]
Here four cases arise
(a) If Ax and Ay are both positive, then
? = tan-1 |Ay / Ax|
(b) If Ax is negative and Ay is positive, then
? = 180º - tan-1 |Ay / Ax|
( c) If Ax is positive and Ay is negative, then
? = 360º - tan-1 |Ay / Ax|
(d) If Ax and Ay are both negative, then
? = 180º + tan-1 |Ay / Ax|
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