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REFLECTION OF LIGHT DEFINITIONS

REFLECTION OF LIGHT
When light rays traveling is a medium reaches the boundary of other medium, they turn back to the first medium. This phenomenon of turning back of light into the same medium after striking the boundary of other medium is called Reflection of Light.

LAWS OF REFLECTION
1. The angle of incident is equal to the angle of reflection i.e. <i = <r 2. The incident ray, the reflected ray and the normal lie on the same plane.

REGULAR REFLECTION
When a beam pass of parallel light rays is incident on a smooth and plane surface, the reflected rays will also be parallel. This type of reflection is called Regular Reflection.

IRREGULAR REFLECTION
When a beam of parallel light rays is scattered in all directions. Therefore the parallel rays incident on the surface will reflect in different directions. This type of reflection is called "Irregular or Diffuse Reflection".

CENTER OF CURVATURE
Center of curvature of a lens or mirror is defined as the center of the sphere of which the less or mirror is a part. C = Center of curvature.

RADIUS OF CURVATURE
Radius of curvature is the radius of sphere of which the lens or mirror is a part.
PC = Radius of curvature OR PC = R

POLE
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The middle or center point of a lens or a mirror is called "Pole" P = Pole.
PRINCIPLE AXIS
The straight line joining the center of curvature to the pole is called Principle Axis. .

PRINCIPLE FOCUS
When a narrow beam of light, parallel to the principle axis and closed to it, is incident on the surface of a mirror or lens, the beam reflected or refracted is converged at a fixed point on the axis. This point is called Principle Axis. F = principle focus.

FOCAL LENGTH
The distance between the pole of a lens or mirror to the principal focus is called Focal Length (PF) of lens or mirror. Focal length is always equal to half of the radius of curvature of lens or mirror. f = R/2.

Write down the characteristics of image formed by a plane mirror
1. Image formed by plane mirror is laterally inverted. This means that right side of the object appears on the left side. 2. Size of image formed by plane mirror is the same as that of size of object. 3. The image formed by plane mirror is virtual because it can not be obtained on the screen. 4. The image is as far behind the mirror as the object is in front of the mirror. Fig.
DEFINE SPHERICAL MIRROR AND IT'S TWO TYPES
SPHERICAL MIRROR
Mirror obtained from a spherical surface is known as Spherical Mirror. A spherical mirror is considered as a section of hollow sphere.

TYPES OF SPHERICAL MIRRORS
There are two types of spherical mirrors. 1. Concave mirror. 2. Convex mirror.
CONCAVE MIRROR
If the inner side of the surface of a spherical mirror is polished to reflect light, the mirror is called a Concave Mirror. Concave mirror converges parallel beam of light.

CONVEX MIRROR
If the outer side of the surface of a spherical mirror is polished to reflect light the mirror is called a Convex Mirror. Convex mirror diverges parallel beam light.

MAGNIFICATION
Magnification of a mirror or lens is defined as the ratio of the size of image to the size of object.
M = height of image/height of object M = hi/ho or M = q/P
REFRACTIVE INDEX
Refractive index is defined as the ratio of sine of the angle of incidence of the sine of the angle of refraction. FORMULA :
m= sine< i/ sine< r
note :Refractive index depends upon the nature of material. It has no unit.
ANGLE OF DEVIATION
The angle at which the light ray is refracted (bend) in a prism is called angle of deviation. It is denoted by < D. Minimum value of angle of deviation is called angle of minimum deviation. It is denoted by <Dm.

SIMPLE PENDULUM

SIMPLE PENDULUM
	simple pendulum consists of a heavy mass particle suspended by a light, flexible and in-extensible string.

	MOTION OF THE BOB OF SIMPLE PENDULUM
	The motion of the bob of simple pendulum simple harmonic motion if it is given small displacement. In order to prove this fact consider a simple pendulum having a bob of mass 'm' and the length of pendulum is 'l'. Assuming that the mass of the string os pendulum is negligible. When the pendulum is at rest at position 'A', the only force acting is its weight and tension in the string. When it is displaced from its mean position to another new position say 'B' and released, it vibrates to and fro around its mean position.
	Suppose that at this instant the bob is at point 'B' as shown below :

	FORCES ACTING ON THE BOB
	1. Weight of the bob (W) acting vertically downward. 2. Tension in the string (T) acting along the string.
	The weight of the bob can be resolved into two rectangular components:
	a. Wcosq along the string. b. Wsinq perpendicular to string.
	Since there is no motion along the string, therefore, the component Wcosq must balance tension (T) i.e. Wcosq = T
	This shows that only Wsinq is the net force which is responsible for the acceleration in the bob of pendulum.
	According to Newton's second law of motion Wsinq will be equal to m x a
	i.e. Wsinq = m a
	Since Wsinq is towards the mean position, therefore, it must have a negative sign. i.e. m a = - Wsinq
	But W = mg
	m a = - mgsinq
	a = - gsinq
	In our assumption q is very small because displacement is small, in this condition we can take sinq = q
	Hence a = - gq ----------- (1)
	If x be the linear displacement of the bob from its mean position, then from figure, the length of arc AB is nearly equal to x
	From elementary geometry we know that:

	Where s= x, r = l
	Putting the value of q in equation (1)

	As the acceleration of the bob of simple pendulum is directly proportional to displacement and is directed towards the mean position, therefore the motion of the bob is simple harmonic when it is given a small displacement

FORCE & MOTION

"Force is an agent which changes or tends to change the state of rest or of uniform motion of a body."
In the light of Newton's 2nd law of motion Force may be defined as :
"Force acting on a body is equal to the product of the mass and acceleration produced in the body."
i.e.
F = ma
Force can accelerate or decelerate a body. Force is a vector quantity.
UNITS OF FORCE
(i) NEWTON (N) in S.I system (ii) DYNE in C.G.S system (iii) POUND (Lb) in BRITISH ENGINEERING SYSTEM (F.P.S)
NEWTON
Newton is the unit of force and can be defined as:
"The amount of force that produces an acceleration of 1 m/s²in a body of mass 1-kg is equal to 1 NEWTON."
1 N = 1 kg x 1m/s²
[ N = kg m/s²]
NEWTON 'S FIRST LAW OF MOTION
STATEMENT: Newton 's first law of motion states that:
"Every body remains at rest or continues to move with uniform velocity in straight line unless an unbalanced force acts upon it".
EXPLANATION
First law of motion consists of two parts:
PART NO 1:
The first part states that a body at rest remains at rest unless an unbalanced force acts upon it. This part is in accordance with our common experience for example, a book lying on a table remains at rest unless it is lifted or pushed by an external force.
PART NO 2 :
Second part states that a body in motion remains in motion with uniform velocity unless an unbalance force acts upon it. This part is not self-evident because a ball pushed once does not continue its motion forever. A little consideration however, shows that there is an opposing force like ground friction and air friction acting in this case. These frictional forces are responsible to stop the ball. If we eliminate these opposing forces, a body in motion will continue its motion forever.
INERTIA
"Tendency of a body by virtue of which the body at rest or moving with uniform velocity retains its state is called INERTIA."
OR
"Property of a body by which a body resists a force, applied to it to change its state of rest or of uniform velocity is called INERTIA."
INERTIA of a body is directly related to its mass. Heavy bodies have greater inertia while lighter bodies have little inertia.
LAW OF INERTIA AND THE FIRST LAW OF MOTION
Every body in the universe opposes the force which tends to change its state of rest or of uniform motion. This property INERTIA is a direct consequence of FIRST LAW OF MOTION. As heavy bodies due to greater INERTIA requires forces of large magnitude and bodies of small masses require small forces. By the above explanation of INERTIA we conclude that the state of rest or motion does not change by its self unless an external force acts upon it, which is according to the FIRST LAW OF MOTION. Thus the FIRST LAW OF MOTION is also called LAW OF INERTIA.

AREA OF PARALLELOGRAM

COMMUTATIVE LAW OF VECTOR ADDITION
Consider two vectors and . Let these two vectors represent two adjacent sides of a parallelogram. We construct a parallelogram
OACB as shown in the diagram. The diagonal OC represents the resultant vector

	From above figure it is clear that:
	This fact is referred to as the commutative law of vectr addition .
	ASSOCIATIVE LAW OF VECTOR ADDITION
	The law states that the sum of vectors remains same irrespective of their order or grouping in which they are arranged. Consider three vectors , and
	Applying "head to tail rule" to obtain the resultant of (+ ) and (+ )
	Then finally again find the resultant of these three vectors :

	This fact is known as the ASSOCIATIVE LAW OF VECTOR ADDITION.

KINEMATICS

KINEMATICS
"Kinematics is the branch of Physics in which we discuss bodies at rest or motion without the reference of external agent that causes motion or rest." OR "The branch of physics which deals with the description of motion of objects without reference to the force or agent causing motion in it, is called Kinematics."
REST
	"If a body does not change its position with respect to its surroundings then the body is said to be in a state of rest."
	MOTION
	"If a body continuously changes its position with respect to its surrounding than it is said to be in a state of motion."
	TYPES OF MOTION
	Motion of objects can be divided into three categories.
	(i) TRANSLATIONAL MOTION (ii) ROTATIONAL MOTION (iii) VIBRATIONAL MOTION
	TRANSLATIONAL MOTION
	"Motion of a body in which every particle of the body is being displaced by the same amount is called Translational Motion".
	EXAMPLE: (i) Motion of a person on a road. (ii) Motion of a car or truck on a road.
	ROTATIONAL MOTION
	"Type of motion in which a body rotates around a fixed point or axis is called Rotational Motion."
	EXAMPLE: (i) Motion of wheel (ii) Motion of the blades of a fan
	VIBRATIONAL MOTION
	"Type of motion in which a body or particle moves to and fro about a fixed point or mean position is called Vibratory Motion."
	EXAMPLE: (i) Motion of simple pendulum (ii) Motion of the wires of guitar (iii) Motion of swing
	DISPLACEMENT
	"Distance between two points in a particular direction is called Displacement." OR Displacement may also be defined as "the minimum distance between two points in a particular direction."
	It is a vector quantity and is always directed from the initial point to the terminal point. It is denoted by "d".
	SPEED
	"Distance covered by a moving body in one second is called its Speed". OR "Distance covered by a body in unit time is called Speed".
	Speed is a scalar quantity.
	FORMULA
	Speed = Distance traveled/Time taken OR v = S/t
	UNIT
	Unit of speed in S.I system is "m/sec".
	VELOCITY
	"Distance covered by a body in a particular direction in one second is called Velocity". OR "Displacement of a body in unit time is called Velocity". OR "Change of position of a body per second in a particular direction is called Velocity."
	FORMULA
	velocity = displacement/time
	UNIT
	In S.I system unit of velocity is meter/second. It is a vector quantity.
	ACCELERATION
	"The rate of change of velocity of a body is called Acceleration." OR "Change in velocity of a body in unit time is called its acceleration."
	It is denoted by "a". It is a vector quantity. If a body moves with uniform velocity or constant velocity then its acceleration will be zero. UNIT: m/sec².
	FORMULA
	Acceleration = change in velocity/time OR a = DV/t

LAW OF CONSERVATION OF ENERGY

LAW OF CONSERVATION OF ENERGY
According to the law of conservation of energy : "Energy can neither be created nor it is destroyed, however energy can be converted from one form energy to any other form of energy"
SHOW THAT THE MOTION OF A SIMPLE PENDULUM IS ACCORDING TO THE LAW OF CONSERVATION ENERGY. OR PROVE THE LAW OF CONSERVATION WITH THE HELP OF A SUITABLE EXAMPLE.
	We know that the motion of the bob of a simple pendulum is simple harmonic motion. Here we have to prove that the energy is conversed during the motion of pendulum. Proof: Consider a simple pendulum as shown in the diagram.

	Energy Conservation At Point ‘A’
	At point ‘A’ velocity of the bob of simple pendulum is zero. Therefore, K.E. at point ‘A’ = 0. Since the bob is at a height (h), Therefore, P.E. of the bob will be maximum. i.e. P.E. = mgh. Energy total = K.E. + P.E Energy total = 0 + mgh Energy total = mgh This shows that at point A total energy is potential energy.
	Energy Conservation At Point ‘M’
	If we release the bob of pendulum from point ‘A’, velocity of bob gradually increases, but the height of bob will decreases from point to the point. At point ‘M’ velocity will become maximum and the height will be nearly equal to zero. Thus , K.E. = maximum = 1/2mV² but P.E. = 0. Energy total = K.E. + P.E Energy total = 1/2mV² + 0 Energy total = 1/2mV² This shows that the P.E. at point is completely converted into K.E. at point ‘M’.
	Energy Conservation At Point ‘B’
	At point M the bob of Pendulum will not stop but due to inertia, the bob will moves toward the point ‘B’. As the bob moves from ‘M’ to ‘B’, its velocity gradually decreases but the height increases. At point ‘B’ velocity of the bob will become zero. Thus K.E. at point ‘B’ = 0 but P.E. = max. P.E. = mgh. Energy total = K.E. + P.E. Energy total = 0 + mgh Energy total = mgh This shows that at point B total energy is again potential energy.
	CONCLUSION
	Above analysis indicates that the total energy during the motion does not change. I.e. the motion of the bob of simple pendulum is according to the law of conservation of energy.

ENERGY

ENERGY
"The ability of a body to perform work is called Energy".
A body cannot perform work if it does not posses energy.A body cannot perform work more than the amount of energy.
	It is a scalar quantity.
	UNITS OF ENERGY
	(i) Joule (ii) Calorie [NOTE: 1 Calorie = 4.2 joule.] (iii) KWatt-Hour
	TYPES OF ENERGY
	There aree numerous types of energy such as: Heat Energy Light Energy Sound Energy Nuclear Energy Chemical Energy Electrical Energy Solar Energy Wind Energy Kinetic Energy Potential Energy etc. etc.
	POWER
	"The rate of work done of a body is called Power".
	AVERAGE POWER
	Average power of a body doing work is numerically equal to the totla work done divided by the time taken to perform the work.
	MATHMATICALLY
	Power = Work done/time Power = Work/t but [work = Fd] therefore Power = Fd/t
	UNITS OF POWER
	(i) watt [1 watt = 1joule/sec ] (ii) Kilo watt [1Kw = 1000 watt] (iii) Mega watt (Mw) [1Mw = 10⁶ watt] (iv) Horse power [1Hp = 746w]
	POTENTIAL ENERGY
	INTRODUCTION
	Energy stored by a body by any means is called "Potential Energy".
	DEFINITION
	"The energy stored by a body due to its position in gravitational field is known as ‘Gravitational Potential Energy’".
	FORMULA
	Consider a body of mass "m" placed at a height of "h" from the surface of earth. Force = Weight = W but displacement (d) = h

	Work done = Fd OR Work done = Wh [but W = mg]
	work done = mgh WE know that the work done in lifting the body is stored in the body in the form of Potential Energy. Thus P.E. = mgh

Conic Section

Trigonometric Functions and their Graphs - Maths

Chapter 11: Trigonometric Functions and their Graphs

Notes (Solutions) of Chapter 11: Trigonometric Functions and their Graphs, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore.

Contents & summary

Introduction
- Domains and Ranges of Sine and Cosine Functions
- Domains and Ranges of Tangent and Cotangent Functions
- Domain and Range of Secant Function
- Domain and Range of Cosecant Function
Period of Trigonometric Functions
- Exercise 11.1
Values of Trigonometric Functions
Graphs of Trigonometric Functions
Graph of y=sinx from −2π to 2π
Graph of y=cosx from −2π to 2π
Graph of y=tanx from −π to π
Graph of y=cotx from −2π to π
Graph of y=secx from −2π to 2π
Graph of y=cscx from −2π to 2π
- Exercise 11.2

FSc Notes