MACHINES

MACHINE
A machine is a device by means of which work can be performed easily or in a convenient manner. A machine can be used :  To lift heavy loads by applying little force.  To enlarge magnitude of force  To increase rate of work done  To change the direction of force Example of simple machines are : Lever, pulley, inclined plane, wedge, screw etc.
EFFORT OR POWER
The power directly applied to a machine to lift a load is called Effort or Power. It is denoted by ‘P’.
LOAD OR WEIGHT
The weight lifted by a machine is called Load. It is denoted by ‘W’.
MECHANICAL ADVANTAGE
The ratio of weight (load) lifted by a machine to the force(effort) applied on a machine is called mechanical advantage of the machine. Greater the value of mechanical advantage of a machine, more easier is the work done. Mathematically,
M.A = load/effort
OR
M.A = W/P
UNIT:
It has no unit.
INPUT
Amount of work done on a machine by a given effort (force) is called input of a machine.
Input = effort x distance through which effort acts OR
input = P x d
OUTPUT
Amount of work done by a machine on the load (weight) is called output of the machine.
Output = load x distance covered by the load OR
Output = W x D
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EFFICIENCY
The ratio of output of a machine to the input of machine is called its efficiency.
h = output/input h = (W x D)/(P x d) Efficiency in %:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         h = (W x D)/(P x d )x100
UNIT:
It has no unit.
IDEAL MACHINE
An ideal machine is a hypothetical machine whose output is equal to its input. For an ideal machine
output = input
Efficiency of an ideal machine is 100% because there is no loss of energy in an ideal machine due to friction or any other means that can waste useful energy.
M.A of an ideal machine is d / h.
LEVER
Lever is a simple machine which is used to lift heavy bodies or heavy load in a very easy way. Lever consists of a rigid bar capable to rotate about a fixed axis called fulcrum. Effort is applied at one end of the bar and weight can be lifted from the other end.
TYPES OF LEVER
There are three kinds of lever depending upon the positions of load , effort and fulcrum.
FIRST KIND OF LEVER
In the first kind of lever, the fulcrum F lies between effort (P) and load (W).
Example: common balance, seesaw, scissors, handle of hand pump.
SECOND KIND OF LEVER
In the second kind of lever, load (W) lies between effort (P) and fulcrum (F).
Example: door, nutcracker, punching machine.
THIRD KIND OF LEVER
In the third kind of lever, effort (P) lies between load (W) and fulcrum (F). Example: forceps, jaws, human forearm, firetong.

CIRCULAR MOTION AND GRAVITATION

GRAVITATION
Every object in our universe attracts the other object with certain fore towards its center. This force of  attraction is known asGRAVITATIONAL FORCE and the phenomenon is called GRAVITATION. This is  gravitational force which is responsible for the uniformity or regularity in our daily astronomical life. The  whole system of the universe is in order only due to this force. Due to gravitation, the system of our  universe is working uniformly and smoothly. The planets around the earth or around the sun moves in an  orderly motion due to gravitation.
NEWTON’S LAW OF GRAVITATION
In order to explain the gravitational force between two bodies, Newton formulated a fundamental law known after his name i.e. "NEWTON'S LAW OF GRAVITATION" Newton’s law of gravitation states that every object in the universe attracts the other object with a force and :
(1) The gravitational force of attraction between two bodies is directly proportional to the product of their  masses.
F a m1 x m2 ------- (1)
(2) The gravitational force of attraction between two bodies is inversely proportional to the square of the  distance between their centers.
F a 1/d2 --------- (2)
MATHEMATICAL REPRESENTATION
Combining (1) and (2)
F a m1m2 /d2
F = G m1m2/d2
Where G = universal gravitational constant
Value of G:
G = 6.67 x 10-11 Nm2/kg2
MASS OF THE EARTH
Consider a body of mass ‘m’ placed on the surface of the earth. Let the mass of the earth is ‘Me’ and  radius  of earth is ‘Re’ .
Gravitational force of attraction between earth and body is
F = G m Me/ Re2
We know that the force of attraction of the earth on a body is equal to weight the weight of body.
i.e
F = W
therefore
W = G m Me/ Re2
But W = mg
mg = G m Me/ Re2
or g = G Me/Re2
or Me = g x Re2/G
From astronomical data:  g= 9.8 m/s2  Re = 6.4 x 106 m  G = 6.67 x 10-11 N-m2/kg2  Putting these values in the above equation.
Me = 9.8 (6.4 x 106)2/6.67 x 10-11 or

STATICS

Statics
Statics is the branch of mechanics which deals with the study of bodies at rest under a number of    forces, the equilibrium, conditions of equilibrium, types of equilibrium and torque etc.
Equilibrium
A body is said to be in equilibrium if it is at rest or moving with uniform velocity.    In other words if the linear and angular acceleration of a body are zero, the body is said to be in    equilibrium.    Or we can say that when two or more forces act on a body such that their resultant or combining effect    on the body is Zero and the body retains its state of rest or of uniform motion then the body is said to    be in equilibrium.
Example
A book lying on the table, suspended bodies, all stationary bodies , jump by using parachute.
Types of equilibrium
With respect to the state of a body, equilibrium may be divided into two categories:    1. Static equilibrium.   2. Dynamic equilibrium.   For latest information , free computer courses and high impact notes visit :
Static equilibrium
If the combined effect of all the forces acting on a body is zero and the body is in the state of rest then    its equilibrium is termed asstatic equilibrium.   For example: All stationary bodies
Dynamic equilibrium
when a body is in state of uniform motion and the resultant of all the forces acting upon it is zero then it    is said to be in dynamic equilibrium.   For example: Jump by using parachute.
Conditions of equilibrium
There are two conditions of equilibrium are as follows
First condition of equilibrium
The first condition of equilibrium stated as follow:
To maintain the transitional equilibrium in a body the vector sum of all the forces acting on the body is    equal to zero    i.e.

MULTIPLICATION & DIVISION OF VECTOR BY A NUMBER (SCALAR)

MULTIPLICATION OF A VECTOR BY A SCALAR
When a vector is multiplied by a positive number (for example 2, 3 ,5, 60 unit etc.) or a scalar only its magnitude is changed but its direction remains the same as that of the original vector. If however a vector is multiplied by a negative number (for example -2, -3 ,-5, -60 unit etc.) or a scalar not only its magnitude is changed but its direction also reversed.
The product of a vector by a scalar quantity (m) follows the following rules:
	(m)  = (m) which is called commutative law of multiplication.
	m(n) = (mn) which is called associative law of multiplication .
	(m + n) = m+ n which is called distributive law of multiplication .
	DIVISION OF A VECTOR BY A SCALAR
	The division of a vector by a scalar number (n) involves the multiplication of the vector by the reciprocal of the number (n) which generates a new vector.
	Let n represents a number or scalar and m is its reciprocal then the new vector  is given by :
	where m = 1/n
	and its magnitude is given by:
	The direction of  is same as that of if (n) is a positive number.  The direction of  is opposite as that of if (n) is a negative number.

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.

Chemical Names

Systematic and Common Names

There are many ways to name a chemical. Here is a look at the difference between different types of chemical names, including systematic names, common names, vernacular names and CAS numbers.

Systematic or IUPAC Name

The systematic name, also called the IUPAC name is the preferred way to name a chemical because each systematic name identifies exactly one chemical. The systematic name is determined by guidelines set forth by the International Union of Pure and Applied Chemistry (IUPAC).

Common Name

A common name is defined by the IUPAC as a name that unambiguously defines a chemical, yet does not follow the current systematic naming convention. An example of a common name is acetone, which has the systematic name 2-propanone.

Vernacular Name

A vernacular name is a name used in a lad, trade or industry that does not unambiguously describe a single chemical. For example, copper sulfate is a vernacular name which may refer to copper(I) sulfate or copper(II) sulfate.

Archaic Name

An archaic name is an older name for a chemical that predates the modern naming conventions. It's helpful to know archaic names of chemicals because older texts may refer to chemicals by these names. Some chemicals are sold under archaic names or may be found in storage labeled with the older names. An example of this is muriatic acid, which is the archaic name for hydrochloric acid and is one of the names under which hydrochloric acid is sold.

CAS Number

A CAS number is an unambiguous identifier assigned to a chemical by the Chemical Abstracts Service (CAS), a part of the American Chemical Society. CAS numbers are assigned sequentially, so you can't tell anything about the chemical by its number. Each CAS number consists of three strings of numbers that are separated by hyphens. The first number contains up to six digits, the second number is two digits, and the third number is a single digit.

Other Chemical Identifiers

Although the chemical names and CAS Number are the most common way to describe a chemical, there are other chemical identifiers you may encounter. Examples include numbers assigned by PubChem, ChemSpider, UNII, EC number, KEGG, ChEBI, ChEMBL, RTES number and the ATC code.

Example of Chemical Names

Putting it all together, here are the names for CuSO4·5H2O:

Systematic (IUPAC) Name: copper(II) sulfate pentahydrate

Common Names: copper(II) sulfate, copper(II) sulphate, cupric sulfate, cupric sulphate

Vernacular Name: copper sulfate, copper sulphate

Archaic Name: blue vitriol, bluestone, copper vitriol

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