DISTRIBUTIVE LAW FOR DOT PRODUCT

DISTRIBUTIVE LAW FOR DOT PRODUCT
According to distributive law for dot product:
	PROOF
	Consider three vectors , and .Here we will use geometric interpretation of dot product by drawing projection as shown below. First we obtain the sum of vectors  and by head to tail rule then we draw projection  and from the terminal point of vector respectively onto the direction of . The dot product  is equal to the projection of vector  onto the direction of  multiplied by the magnitude of. i.e.

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

RESOLUTION OF VECTOR

DEFINITION
The process of splitting a vector into various parts or components is called "RESOLUTION OF VECTOR"
These parts of a vector may act in different directions and are called "components of vector".
	We can resolve a vector into a number of components .Generally there are three components of vector viz. Component along X-axis called x-component Component along Y-axis called Y-component Component along Z-axis called Z-component
	Here we will discuss only two components x-component & Y-component which are perpendicular to each other.These components are called rectangular components of vector.
	METHOD OF RESOLVING A VECTOR INTO RECTANGULAR COMPONENTS
	Consider a vector  acting at a point making an angle q with positive X-axis. Vector  is represented by a line OA.From point A draw a perpendicular AB on X-axis.Suppose OB and BA represents two vectors.Vector OA is parallel to X-axis and vector BA is parallel to Y-axis.Magnitude of these vectors are Vx and Vy respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector.Thus Vx and Vy are the rectangular components of vector . Vx = Horizontal component of . Vy = Vertical component of .
	MAGNITUDE OF HORIZONTAL COMPONENT
	Consider right angled triangle DOAB
	MAGNITUDE OF VERTICAL COMPONENT
	Consider right angled triangle DOAB

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

Nomenclature of Simple Alkene Chain Molecules

Simple Alkene Chains

An alkene is a molecule made up entirely of carbon and hydrogen where on or more carbon atoms are connected by double bonds. The general formula for an alkene is CnH2n where n is the number of carbon atoms in the molecule.

Alkanes are named by adding the the -ene suffix to the prefix associated with the number of carbon atoms present in the molecule. A number and dash before the name denotes the number of the carbon atom in the chain that begins the double bond.
For example: 1-hexene is a six carbon chain where the double bond is between the first and second carbon atoms.

Click image to enlarge the molecule.

Ethene

Todd Helmenstine

Number of Carbons: 2
Prefix: eth- Number of Hydrogens: 2(2) = 4
Molecular Formula: C2H4

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Propene

Todd Helmenstine

Number of Carbons: 3
Prefix: prop- Number of Hydrogens: 2(3)= 6
Molecular Formula: C3H6

Butene

Todd Helmenstine

Number of Carbons: 4
Prefix: but- Number of Hydrogens: 2(4) = 8
Molecular Formula: C4H8

Pentene

Todd Helmenstine

Number of Carbons: 5
Prefix: pent- Number of Hydrogens: 2(5) = 10
Molecular Formula: C5H10

Hexene

Todd Helmenstine

Number of Carbons: 6
Prefix: hex- Number of Hydrogens: 2(6)= 12
Molecular Formula: C6H12

Heptene

Todd Helmenstine

Number of Carbons: 7
Prefix: hept- Number of Hydrogens: 2(7) = 14
Molecular Formula: C7H14

Octene

Todd Helmenstine

Number of Carbons: 8
Prefix: oct- Number of Hydrogens: 2(8) = 16
Molecular Formula: C8H16

Nonene

Todd Helmenstine

Number of Carbons: 9
Prefix: non- Number of Hydrogens: 2(9) = 18
Molecular Formula: C9H18

Decene

Todd Helmenstine

Number of Carbons: 10
Prefix: dec- Number of Hydrogens: 2(10) = 20
Molecular Formula: C10H20

Isomer Numbering Scheme

These three structures illustrate the numbering scheme for isomers of alkene chains. The carbon atoms are numbered from left to right. The number represents the location of the first carbon atom that is part of the double bond.
In this example: 1-hexene has the double bond between carbon 1 and carbon 2, 2-hexene between carbon 2 and 3, and 3-hexene between carbon 3 and carbon 4.
4-hexene is identical to 2-hexene and 5-hexene is identical to 1-hexene. In these cases, the carbon atoms would be numbered from right to left so the lowest number would be used to represent the molecule's name.