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Scalars & Vectors,Motion,Motion in two Dimension,Torque, Angular, Momentum & Equilibrium,Gravitation,Work, Power & Energy
Posted by altaf
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Recent Posts
- Scalars and Vectors
- Motion
- Motion in two Dimension
- Torque, Angular, Momentum and Equilibrium
- Gravitation
- Work, Power and Energy
Scalars and Vectors
Scalars
Physical quantities which can be completely specified by1. 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 by1. 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
DefinitionSplitting 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
Fx / F = Cos θ => Fx = F cos θ
Fy / F = sin θ => Fy = F sin θ
Consider two vectors A1 and A2 making angles θ1 and θ2 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.
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 θ1 + A2 cos θ2 ................. (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 θ1 + A2 sin θ2 ................. (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| = √ (Ax)2 + (Ay)2 .................. (3)
Step 4
By applying the trigonometric ratio of tangent θ on triangle OAB, we can find the direction of the resultant vector A i.e. angle θ which A makes with the positive x-axis.
tan θ = Ay / Ax
θ = tan-1 [Ay / Ax]
Here four cases arise
(a) If Ax and Ay are both positive, then
θ = tan-1 |Ay / Ax|
(b) If Ax is negative and Ay is positive, then
θ = 180º - tan-1 |Ay / Ax|
(c) If Ax is positive and Ay is negative, then
θ = 360º - tan-1 |Ay / Ax|
(d) If Ax and Ay are both negative, then
θ = 180º + tan-1 |Ay / Ax|
Addition of Vectors by Law of Parallelogram
According to the law of parallelogram of addition of vectors, if we are given two vectors. A1 and A2 starting at a common point O, represented by OA and OB respectively in figure, then their resultant is represented by OC, where OC is the diagonal of the parallelogram having OA and OB as its adjacent sides.R = A1 + A2
Or
OC = OA + OB
But OB = AC
Therefore,
OC = OA + AC
β is the angle opposites to the resultant.
Magnitude of the resultant can be determined by using the law of cosines.
R = |R| = √A1(2) + A2(2) - 2 A1 A2 cos β
Direction of R can be determined by using the Law of sines.
A1 / sin γ = A2 / sin α = R / sin β
This completely determines the resultant vector R.
Properties of Vector Addition
1. Commutative Law of Vector Addition (A+B = B+A)
Consider two vectors A and B as shown in figure. From figure
OA + AC = OC
Or
A + B = R .................... (1)
And
OB + BC = OC
Or
B + A = R ..................... (2)
Since A + B and B + A, both equal to R, therefore
A + B = B + A
Therefore, vector addition is commutative.
2. Associative Law of Vector Addition (A + B) + C = A + (B + C)
Consider three vectors A, B and C as shown in figure. From figure using head - to - tail rule.
OQ + QS = OS
Or
(A + B) + C = R
And
OP + PS = OS
Or
A + (B + C) = R
Hence
(A + B) + C = A + (B + C)
Therefore, vector addition is associative.
Product of Two Vectors
1. Scalar Product (Dot Product)
2. Vector Product (Cross Product)
1. Scalar Product OR Dot Product
If the product of two vectors is a scalar quantity, then the product itself is known as Scalar Product or Dot Product.
The dot product of two vectors A and B having angle θ between them may be defined as the product of magnitudes of A and B and the cosine of the angle θ.
A . B = |A| |B| cos θ
A . B = A B cos θ
The scalar product of vector A and vector B is equal to the magnitude, A, of vector A times the projection of vector B onto the direction of A.
If B(A) is the projection of vector B onto the direction of A, then according to the definition of dot product.
A . B = A B cos θ {since B(A) = B cos θ}
Examples of dot product are
W = F . d
P = F . V
Commutative Law for Dot Product (A.B = B.A)
If the order of two vectors are changed then it will not affect the dot product. This law is known as commutative law for dot product.
A . B = B . A
if A and B are two vectors having an angle θ between then, then their dot product A.B is the product of magnitude of A, A, and the projection of vector B onto the direction of vector i.e., B(A).
And B.A is the product of magnitude of B, B, and the projection of vector A onto the direction vector B i.e. A(B).
In Δ PQR,
cos θ = A(B) / A => A(B) = A cos θ
In Δ ABC,
cos θ = B(A) / B => B(A) = B cos θ
Therefore,
A . B = A B(A) = A B cos θ
B . A = B A (B) = B A cos θ
A B cos θ = B A cos θ
A . B = B . A
Thus scalar product is commutative.
Distributive Law for Dot Product
A . (B + C) = A . B + A . C
Consider three vectors A, B and C.
B(A) = Projection of B on A
C(A) = Projection of C on A
(B + C)A = Projection of (B + C) on A
Therefore
A . (B + C) = A [(B + C}A] {since A . B = A B(A)}
= A [B(A) + C(A)] {since (B + C)A = B(A) + C(A)}
= A B(A) + A C(A)
= A . B + A . C
Therefore,
B(A) = B cos θ => A B(A) = A B cos θ1 = A . B
And C(A) = C cos θ => A C(A) = A C cos θ2 = A . C
Thus dot product obeys distributive law.
2. Vector Product OR Cross Product
When the product of two vectors is another vector perpendicular to the plane formed by the multiplying vectors, the product is then called vector or cross product.
The cross product of two vector A and B having angle θ between them may be defined as "the product of magnitude of A and B and the sine of the angle θ, such that the product vector has a direction perpendicular to the plane containing A and B and points in the direction in which right handed screw advances when it is rotated from A to B through smaller angle between the positive direction of A and B".
A x B = |A| |B| sin θ u
Where u is the unit vector perpendicular to the plane containing A and B and points in the direction in which right handed screw advances when it is rotated from A to B through smaller angle between the positive direction of A and B.
Examples of vector products are
(a) The moment M of a force about a point O is defined as
M = R x F
Where R is a vector joining the point O to the initial point of F.
(b) Force experienced F by an electric charge q which is moving with velocity V in a magnetic field B
F = q (V x B)
Physical Interpretation of Vector OR Cross Product
Area of Parallelogram = |A x B|
Area of Triangle = 1/2 |A x B|
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Motion
DefinitionIf an object continuously changes its position with respect to its surrounding, then it is said to be in state of motion.
Rectilinear Motion
The motion along a straight line is called rectilinear motion.
Velocity
Velocity may be defined as the change of displacement of a body with respect time.Velocity = change of displacement / time
Velocity is a vector quantity and its unit in S.I system is meter per second (m/sec).
Average Velocity
Average velocity of a body i...
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Motion in two Dimension
Projectile Motion
A body moving horizontally as well as vertically under the action of gravity simultaneously is called a projectile. The motion of projectile is called projectile motion. The path followed by a projectile is called its trajectory.Examples of projectile motion are
1. Kicked or thrown balls
2. Jumping animals
3. A bomb released from a bomber plane
4. A shell of a gun.
Analysis of Projectile Motion
Let us consider a body of mass m, projected an angle θ with the horizonta...
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Torque, Angular, Momentum and Equilibrium
Torque or Moment of Force
DefinitionIf a body is capable of rotating about an axis, then force applied properly on this body will rotate it about the axis (axis of rotating). This turning effect of the force about the axis of rotation is called torque.
Torque is the physical quantity which produces angular acceleration in the body.
Explanation
Consider a body which can rotate about O (axis of rotation). A force F acts on point P whose position vector w.r.t O is r.
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Gravitation
The property of all objects in the universe which carry mass, by virtue of which they attract one another, is called Gravitation.Centripetal Acceleration of the Moon
Newton, after determining the centripetal acceleration of the moon, formulated the law of universal gravitation.Suppose that the moon is orbiting around the earth in a circular orbit.
If V = velocity of the moon in its orbit,
Rm = distance between the centres of earth and moon,
T = time taken by the moon to c...
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Work, Power and Energy
Work
Work is said to be done when a force causes a displacement to a body on which it acts.Work is a scalar quantity. It is the dot product of applied force F and displacement d.
W = F . d
W = F d cos θ .............................. (1)
Where θ is the angle between F and d.
Equation (1) can be written as
W = (F cos θ) d
i.e., work done is the product of the component of force (F cos θ) in the direction of displacement and the magnitude of displacement d.
equ...
Physics Notes for Metric (x) Class (Karachi Board)
Posted by altaf
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Physics Notes for Metric Class (Karachi Board)
Dear Students, you are welcome to Study Guide Centre. We are presenting the Notes for Metric class, Karachi Board.Here are the notes for Physics.
Metric Class:
Chapter # 1: Introduction
Chapter # 2: Measurments
Chapter #3 : Scalars and Vectors
Chapter #4 : Kinematics
Chapter #5 : Force and Motion
Chapter # 6: Statics
Chapter # 7: Circular Motion and Gravitation
Chapter # 8: Work Energy and Power
Chapter # 9: Machines
Chapter # 10: Matter
Chapter # 11: Heat
Chapter # 12: Waves and Sounds
Chapter # 13: Propagation and Reflection of Light
Chapter # 14: Reflection of Light and Optical Instruments
Chapter # 15: Nature of Light and Electromagnetics spectrum
Chapter # 16: Electricity
Chapter # 17: Magnetic and Electromagnetism
Chapter # 18: Electronics
Chapter # 19: Nuclear Physics