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  • INTRODUCTION TO EUCLID’S GEOMETRY | Study

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    • FRICTION | Study

      Mind Map Overal Idea Content Speed Notes Quick Coverage Friction: Friction is the force that resists the motion of one object against another.  It is the force felt between two surfaces when one attempts to slide against the other. Causes of Friction: Friction exists between two surfaces due to irregularities on the surfaces of the objects readmore

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      Friction:

      Friction is the force that resists the motion of one object against another.  It is the force felt between two surfaces when one attempts to slide against the other.

      Causes of Friction: Friction exists between two surfaces due to irregularities on the surfaces of the objects in contact, interlocking of micro-level irregularities of the two surfaces and ploughing of harder surfaces into smoother surfaces.

      Factors Affecting Friction:

      (i) Roughness of the surface.

      (ii) The extent to which the two surfaces press against each other.

      (iii) Nature of the surface· (Scroll down till end of the page)

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      Types of Friction: Friction is classified into three types as: Static Friction, Sliding Friction, Rolling Friction.
      (i) Static Friction: When a body is at rest, the force of friction is called the static friction and is always equal and opposite to the applied force.

      limiting friction: The force of friction which acts when the body is just at the verge of sliding on the surface is called limiting friction.

      Motion between two contacting surfaces

      (ii) Sliding friction: The friction force which opposes the actual relative sliding motion between two contact surfaces.

      Sliding friction is smaller than static friction.

      (iii) Rolling Friction: The frictional force that exists between two surfaces when a body rolls over the other.

      Rolling friction is smaller than sliding friction.

      Static friction > Sliding friction > Rolling friction

      Effects of Friction

      (i) Friction produces heat.

      (ii) Friction causes wear and tear.

      (iii) Friction opposes motion.

      Advantages of Friction

      (i) Friction between pen and paper enables us to write on the paper.

      (ii) Friction between our feet and the ground allows our movements like standing, walking and running.

      (iii) Friction between the surface of the road and tyres of our vehicles allow the vehicles to move without slipping.

      Disadvantages of Friction:

      (i) Friction causes moving objects to stop or slow down.

      (ii) Friction produces heat causing wastage of energy in machines.

      (iii) Friction causes wear and tear of moving parts of machinery, soles of shoes, etc.

      Friction is a necessary Evil:

      As friction is advantageous to us it is considered as a friend but, due to its disadvantages it is a foe.

      Depending on the circumstance, friction can be a help or a hindrance.

      Thus it is a necessary evil.

      Increasing Friction: By pressing the surfaces together more strongly.

      For example, when brakes are applied on a bicycle or car, the brake pads press against a moving part of the wheel and the force of friction increases.

      Friction can be increased by increasing the roughness of the surfaces in contact.

      For example, treading of shoes and tyres is done to increase friction.

      Reducing Friction: Friction between the sliding surfaces of two objects can be reduced by making the surfaces in contact smooth by polishing them.

      Sliding friction between the moving parts of vehicles and machinery can be reduced by introducing oil, grease, graphite or any other lubricant.

      Lubricant: A lubricant is a substance that reduces friction between surfaces in contact. This reduces the heat generated when the surfaces move. Lubricants can also transmit forces, transport foreign particles, or heat or cool the surfaces. 

      Rolling friction is less than sliding friction. Hence, sliding friction is replaced by rolling friction by using rollers, like ball bearings between the hub and the axles in the moving parts of machines and vehicles.

      Friction is reduced by providing wheels,

      Examples: suitcase, school bags of kids, etc.

      Streamlined shape: Aeroplanes, boats, fishes and birds which move through fluids have bodies of special shape, called streamlined shape, The streamlined shape reduces the friction due to fluid and reduces the energy usage.

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      • Heridity | Study

        Mind Map Overal Idea Content Speed Notes Quick Coverage Content Study Tools Audio, Visual & Digital Content Chapter 9 -Heredity and Evolution 1. Accumulation of variation during Reproduction. Variations in an individual may be an advantage or disadvantage for it. It may enable or disable it to cope with changes in the environment. Advantageous variations readmore

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        • INTRODUCTION TO EUCLID’S GEOMETRY | Study

          Mind Map Overal Idea Content Speed Notes Quick Coverage Introduction to Euclid Geometry The necessity of geometry had been felt from ancient times in different parts of the world. The practical problems faced by people of ancient civilization had developed this branch of mathematics. Let us cite few examples. With floods in the river, the readmore

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          Introduction to Euclid Geometry The necessity of geometry had been felt from ancient times in different parts of the world. The practical problems faced by people of ancient civilization had developed this branch of mathematics.

          Let us cite few examples.

          With floods in the river, the demarcations of land owners on the river-side land were used to wipe out (Scroll down till end of the page)

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          In order to redraw the boundaries, the idea of area was introduced, the idea of area was introduced.

          The volumes of granaries could be measured by using geometry.

          The existence of Egyptian pyramids indicates the use of geometry from olden times.

          In Vedic period, there was a manual of geometrical construction, known as Sulbasutra’s.

          Different geometrical shapes were constructed as altars to perform various Vedic rites.

          The word Geometry originates from the green word ‘Geo’ (earth) and metrein (to measure) Through Geometry was developed and applied from ancient time in various part the world, it was not presented in a systematic manner.

          Later in 300 BC, the Egyptian mathematician Euclid, collected all the known work and arranged it in a systematic manner.

          ‘Elements’ is a classic treatise in geometry which was written by Euclid.

          This was the most influential book. The ‘element’ was used as a text book for several years in western Europe.

          The ‘elements’ started with 28 definitions, five postulates and five common notions and systematically built the rest of plane and solid geometry.

          The geometrical approach given by Euclid is known as Euclid method.

          The Euclid method consists of making a small set of assumptions and then proving many other proposition from these assumptions.

          The assumptions, made were obvious universal truth. The two types of assumption, made were ‘axioms’ and ‘postulates’.

          Euclid’s Definitions Euclid listed 23 definitions in book 1 of the ‘elements’.

          We list a few of them: 1) 2) 3) 4) 5) 6) 7) A point is that which has no part A line is a breadth less length The ends of a line are points A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only.

          The edges of a surface are lines A plane surface is surface which lies evenly with straight lines on its self. Starting with these definitions, Euclid assumed certain assumptions, known as axioms and postulates.

          Euclid’s Axioms Axioms were assumptions which were used throughout mathematics and are not specifically linked to geometry.

          Few of Euclid’s axioms are:

          1) Things which are equal to the same thing are equal to one another.

          2) It equals are added to equals; the wholes are equal.

          3) 4) 5) 6) 7) If equals are subtracted from equals, the remainders are equal.

          Things which coincide one another are equal to one another.

          The whole is greater than the part Things which are double of the same thing are equal to one another.

          Things which are half of the same things are equal to one another.

          All these axioms refer to magnitude of same kind.

          Axiom – 1 can be written as follows: If x = Z and y = Z, then x = y

          Axiom – 2 explains the following: If x = y, then x + Z = y + Z According to axiom – 3, If x = y, then x – Z = y – Z Axiom – 4 justifies the principle of superposition that every thing equals itself.

          Axiom – 5, gives us the concept of comparison. If x is a part of y, then there is a quantity Z such that x = y + Z or x > y Note that magnitudes of the same kind can be added, subtracted or compared.

          Euclid’s Postulates Euclid used the term postulate for the assumptions that were specific to geometry. Euclid’s five postulates are as follows: Postulate 1: A straight line may be drawn from any one point to any other point. Same may be stated as axiom 5.1 Given two distinct points, there is a unique line that passes through them.

          Postulate 2: A terminated line can be produced indefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one another. Postulate 5: If a straight line falling on two straight lines makes the interior angle on the same side of it taken together less than two right angles, then two straight lines, if produced indefinitely, meet on that side on which the sum of the angles is less than two right angles. Postulates 1 to Postulates 4 are very simple and obvious and therefore they are taken a ‘self evident truths’. Postulates 5 is complex and it needs to be discussed. Suppose the line XY falls on two lines AB and CD such that ∠1 + ∠2 < 180°, then the lines AB and CD will intersect at a point. In the given figure, they intersect on left side of PQ, if both are produced. Note: In mathematics the words axiom and postulate may be used interchangeably, though they have distinct meaning according to Euclid. System of Consistent Axioms A system of axioms is said to be consistent, if it is impossible to deduce a statement from these axioms, which contradicts any of the given axioms or proposition. Proposition or Theorem The statement or results which were proved by using Euclid’s axioms and postulates are called propositions or Theorems. Theorem: Two distinct lines cannot have more than one point in common. Proof: Given: AB and CD are two lines. To prove: They intersect at one point or they do not intersect. Proof: Suppose the lines AB and CD intersect at two points P and Q. This implies the line AB passes through the points P and Q. Also the line CD passes through the points P and Q. This implies there are two lines which pass through two distinct point P and Q. But we know that one and only one line can pass through two distinct points. This axiom contradicts out assumption that two distinct lines can have more than one point in common. The lines AB and CD cannot pass through two distinct point P and Q. Equivalent Versions of Euclid’s Fifth Postulate The two different version of fifth postulate a) For every line l and for every point P not lying on l, there exist a unique line m passing through P and parallel to l. b) Two distinct intersecting lines cannot be parallel to the same line.

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          • Light – Reflection and Refraction | Assess

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            • Mensuration | Assess

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              • Mensuration | Study

                Mind Map Overal Idea Content Speed Notes Quick Coverage Perimeter: Length of boundary of a simple closed figure. Perimeter of Rectangle = 2(l +b)Perimeter of Square = 4aPerimeter of Parallelogram = 2(sum of two adjacent sides)Area: The measure of region enclosed in a simple closed figure. Area of a trapezium = half of the sum readmore

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                Perimeter: Length of boundary of a simple closed figure.

                Perimeter of Rectangle = 2(l +b)
                Perimeter of Square = 4a
                Perimeter of Parallelogram = 2(sum of two adjacent sides)
                Area: The measure of region enclosed in a simple closed figure.

                Area of a trapezium = half of the sum of the lengths of parallel sides × perpendiculardistance between them.

                Area of a rhombus = half the product of its diagonals. (Scroll down till end of the page)

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                • MOTION | Study

                  Mind Map Overal Idea Content Speed Notes Quick Coverage Rest And Motion The terms Rest and Motion are relative. Motion: An object is said to be in motion when its position changes with time. Rest: An object is said to be at rest when its position does not change with respect to a reference point readmore

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                  Rest And Motion

                  The terms Rest and Motion are relative.

                  Motion: An object is said to be in motion when its position changes with time.

                  Rest: An object is said to be at rest when its position does not change with respect to a reference point with time.

                  A specific point with respect to which we describe the location of an object is called a reference point. (Scroll down till end of the page)

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                  Distance and Displacement

                  • Distance: The total length of path covered by an object is said to be the distance travelled by it.
                  • Displacement: Gap between the initial and final positions of an object is said to be its displacement. Or
                  • The length of a line segment that joins the initial and final positions of an object is known as the displacement.

                  Difference Between Displacement and Displacement

                  DistanceDisplacement
                  Distance is defined as the total length of the path travelled by an object to go from one point to another.Displacement is defined as the  length of the line segment that joins the initial and final positions of an object.
                  Since distance has only magnitude and its direction cannot be specified always, it  is a scalar quantity.Since displacement has magnitude and it  is specified in a direction  from initial position to final position, it is a vector quantity.
                  Distance can only have positive values.Displacement can have both positive and negative values.
                  Distance depends on the length of the path travelled.Displacement depends only on the initial and final point regardless of the path travelled.
                  Difference Between Displacement and Displacement

                  Speed And Velocity Speed

                  • Speed: The distance travelled by an object in unit time is referred to as speed. 
                  • Its S.I unit is m/s. 
                  • In general speed refers to average speed.
                  • Average speed: For non-uniform motion, the average speed of an object is obtained by dividing the total distance travelled by an object by the total time taken.

                  motion-cbse-notes-class-9-science-1

                  • For a uniform motion, the average speed of an object is equal to its instantaneous speed throughout the path.

                  Velocity

                  • Average Velocity or Velocity : For a uniform motion in a straight path, the average velocity is equal to its instantaneous velocity throughout the path.


                  motion-cbse-notes-class-9-science-2

                  • Velocity of an object is equal to the instantaneous velocity of an object.

                  Differences Between Speed and Velocity

                                    SPEED                VELOCITY
                  It is defined as the rate of change of distance.It is defined as the rate of change of net displacement.
                  It is a scalar quantity.It is a vector quantity.
                  It can never be negative or zero.It can be negative,zero or positive.
                  Speed is velocity without direction.Velocity is directed speed.
                  Speed may or may not be equal to velocity.A body may possess different velocities but the same speed.
                  Speed never decreases with time. For a moving body,Velocity can decrease with time. For a moving body , it can be zero.
                  Speed is never zero.Velocity can be zero.
                  Speed in SI is measured in ms-1Velocity in SI, is measured in ms-1
                  Differences Between Speed and Velocity

                  Uniform And Non-Uniform motion

                  • Uniform motion or non accelerated motion: When an object covers equal distances in equal intervals of time, it is said to be in uniform motion. Uniform motion is a non-accelerated motion.
                  • Non-uniform motion or accelerated motion: Motions where objects cover unequal distances in equal intervals of time. Uniform motion is an accelerated motion.

                  Acceleration

                  Acceleration: Change in the velocity of an object per unit time.


                  motion-cbse-notes-class-9-science-3

                  Graphical representation of motions

                  (i) Distance-time graph

                  For a distance-time graph, time is taken on x-axis and distance is taken on the y-axis.


                  motion-cbse-notes-class-9-science-4

                  [Note: All independent quantities are taken along the x-axis and dependent quantities are taken along the y-axis.]


                  motion-cbse-notes-class-9-science-5

                  (ii) Velocity-time graph

                  Equation of motion by graphical methods

                  Derivation Of Equations Of Motion

                  Equations of motion can be derived by two methods. They are (i) Graphical Method. (ii) Algebraic Method

                  Derivation of The Equations of Motion By Algebraic Method:

                  (a) Velocity-time relation:


                  motion-cbse-notes-class-9-science-6


                  motion-cbse-notes-class-9-science-8

                  Derivation of S = ut + ½ at2

                  (ii) The equation for position-time relation:

                  motion-cbse-notes-class-9-science-9
                  motion-cbse-notes-class-9-science-10motion-cbse-notes-class-9-science-7

                  Derivation of v2 – u2  = 2as

                  (iii) Equation for position-velocity relation:


                  motion-cbse-notes-class-9-science-11

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                  Conclusions From a Distance – Time Graph



                  Uniform Circular Motion

                  When a body moves in a circular path with uniform speed, its motion is called uniform circular motion.

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