Archive for
December, 2009
- Posted by Bertha in Grammar |
- December 23rd, 2009 |
Interjections are words or sounds thrown into a sentence in order to express some feelings of mind.
Hurrah! we have won the match.
Heigh-ho! this is a very hot day.
Fie, fie! you should not make such mistakes.
Wow! what a nice dress.
Hush! I have missed the train.
Pooh! I am not afraid of snake.
Alas! he is so sick.
Oh! what a pleasant surprise.
Ah! it was really sad.
- Posted by Bertha in Grammar |
- December 23rd, 2009 |
Conjunctions are used for joining either one word to another word; or one sentence to another sentence.
One Word to Another Word
When a word is joined with another word by a Conjunction; each of the words is generally of the similar parts of speech; such as,
¨ A noun is joined to another noun or pronoun; an adjective to another adjective; a preposition to another preposition; a verb to another verb.
Noun to Noun – Cows and donkeys are both very helpful animals.
Noun to Pronoun – Sam and you have to come for lunch today.
Pronoun to Pronoun – I and you are both right.
Adjective to Adjective – I am upset, but hopeful.
Preposition to Preposition – A bird flies in and through the air.
Verb to Verb – She came and talked to me wisely.
Noun to Noun – She is a fool as well as a dishonest.
Noun to Noun – Is this baby a boy or a girl?
¨ There are few Conjunctions that often go in pairs; such as, either – or; neither – nor; but – also; both – and; but – also; not only – but also.
Noun to Noun – She is neither an unfair nor a dishonest.
Noun to Noun – He is both an unfair and a dishonest.
Adjective to Adjective – She is both wise and talented.
Adjective to Adjective – She is not only wise, but also talented.
Verb to Verb – You should either work or leave.
Adverb to Adverb – You behaved neither kindly nor wisely.
One Sentence to Another Sentence
Among the Conjunctions that join one sentence to another sentence the mostly can be noticed from the below-mentioned examples: -
First Sentence |
Conjunction |
Second Sentence |
You said |
that |
this pen is mine |
I trust him |
because |
he never tells a lie |
He will come to the party |
if |
he is permitted to come |
I want to know |
whether |
I should go |
I must do this |
unless |
I am stopped by my work |
I must leave now |
since or as |
the rain has stopped |
You must leave your bed |
when |
the sun rises |
Nobody could find out |
where |
the goat was lying hid |
The girl is unwise |
but |
the boy is very wise |
I want to know |
how |
your grandpa is today |
Your cow is older |
than |
mine |
It’s long time |
since |
I last met you |
I left the office |
as soon as |
the rain ceased |
He was so badly injured |
that |
he needed to be hospitalized |
She could not pass the exam |
though |
she tried hard |
I closed the door |
after |
my guests had gone |
The boy is smart |
and |
he is very good in studies |
They didn’t disclose |
why |
they didn’t come |
The mouse will play |
while |
the cat is gone |
- Posted by Bertha in Grammar |
- December 19th, 2009 |
Verb and Subject
Verb is that Parts of Speech by means of which an individual or a thing can be said to do something or to be or become something; or to suffer something. Verbs basically indicate some type of action. In other words, by mans of a Verb we can say something about an individual or a thing.
The word or the group of words indicating the thing or the individual is the Subject of the Verb.
We can find out the Subject of a Verb by asking, What is the thing or Who is the person that is, or suffers, or does?
1) Whale is a fish that does not lay eggs.
2) Sam saw tigers in the zoo.
3) The garden is dug by a gardener.
In the first sentence, what thing is told to be a fish that does not lay eggs? A whale. Therefore, the noun whale is the Subject of the Verb ‘is’.
In the second sentence, what person is told to have seen tigers in the zoo? The person Sam. Therefore, the noun Sam is the Subject of the Verb ‘saw’.
In the third sentence, what thing is told to be dug by a gardener? The ground. Therefore, the noun ground is the Subject of the Verb ‘is dug’.
Whenever, the Verb is associated with the subject a Sentence or the main part of a Sentence is formed.
Verb and Object
If we say, “A cat sleeps”, the action indicated by the Verb sleeps ends with the cat. However, if we say, “A cat killed the mouse”, the action meant by killed does not end with the cat, but it passes on to the mouse who is killed.
1) A cat killed the mouse.
In this sentence, mouse is the Object to the Verb killed.
2) A cat sleeps the mouse.
The sentence makes no sense. The Verb sleeps can’t have an Object after it.
Verbs can be divided into three classes:-
- Transitive
- Intransitive
- Auxiliary
Transitive Verb: – In case of a Transitive Verb, the action does not end with the doer; but the action passes from the doer to the Object. In other words, a Verb that requires an object is a Transitive Verb. For example,
1) Sam wrote a poem.
2) I don’t know whether you have come.
In the first example, ‘poem’ is the Object to the Verb ‘wrote’.
In the second example, ‘whether you have come’ is the Object to the Verb ‘know’.
Intransitive Verb: – In case of an Intransitive Verb, the action ends with the doer, instead passes to the Object from the doer. For example,
- We sleep for being fit and healthy.
Sleep what? Has no answer; this is completely non-sense. Objects can’t come after Verbs like sleep. Therefore, sleep is an Intransitive Verb.
Auxiliary Verb: – An Auxiliary Verb helps to form a tense or a mood of another verb; like,
- Did you learn?
- I have slept.
- Does he know?
- We shall overcome.
- You will go.
N.B. – The Verb which is helped by an Auxiliary Verb is called a Principal Verb.
There are few Verbs that without any alteration of form, can be Intransitive or Transitive as per the sense; like
Intransitive |
Transitive |
Let me wait a bit. |
Don’t wait for me. |
The day breaks at five. |
He breaks the stone with a strike of hammer. |
She burnt with anger. |
The fire burnt up the forest. |
Office starts at ten o’clock. |
They started their trip yesterday. |
The rat steals into the hole. |
The rat steals food. |
The shirt is hanging up. |
Sam is hanging up his shirt. |
I doubted about the fact. |
I doubted the truth of your word. |
Let’s bathe in the sea. |
I bathed my puppy with cold water. |
Owls hide in the day. |
Notorious people hide their faults. |
There are some Verbs that have one form for the Intransitive Verb and another form for the Transitive Verb.
Intransitive |
Transitive |
The sun rises in the east. |
He can’t raise the heavy suitcase. |
You should not sit there. |
I set the things in order. |
You did not fare well. |
I didn’t ferry him across. |
The opponents quailed. |
He quelled his enemy. |
Verb that can not be used in all tenses or moods is known as Defective Verb.
- Posted by Max in Heat |
- December 18th, 2009 |
Nature of heat: Heat is a form of energy. When you give heat to a body, the molecules of the latter become more energetic and move about more rapidly thus increasing their kinetic energy. The body becomes hot.
Temperature: It is the degree of hotness or coldness of a body; the hotter the body, the higher its temperature. Thus it is different from heat; it is only one of its effects.
Units of heat and temperature: In the S. I. system, the unit of heat is the joule. In the C.G>S> system, it is the calorie. One calorie = 4.2 joules. The corresponding units of temperature are the degree Kelvin (k) in the S.I. system and degree Celsius ^{0}C (degree centigrade) in the C.G.S. system. However, the values of a degree is the same i.e. 1 ^{0}C = 1 K. In the Fahrenheit scale, the unit of temperature is the degree Fahrenheit (^{0}F). The relation between ^{0} C, K and ^{0}F is given in the next section.
Thermometers: These are instruments for measuring temperature. Two of the common forms used for elementary study are the Celsius and the Fahrenheit thermometers. Both use mercury as the thermometric substance and both work on the principle that mercury expands equally for equal rises of temperature. In the Celsius thermometer, the two fixed points are 0^{0} C and 100 ^{0} C. The space between the two fixed points is divided into 100 equal parts, each part being a degree Celsius. In the Fahrenheit thermometer, the two fixed points are 32 ^{0} F and 212 ^{0} F. The space between is divided into 180 equal parts, each part forming a degree Fahrenheit (^{0} F).
Conversion from one scale to another:
To convert the ^{0}C to ^{0}F and vice versa, we can use the formula
F = C × 9 / 5 + 32
This is easy to understand for 100 Celsius divisions occupy the whole space that is occupied by 180 Fahrenheit divisions. So 1^{0} C division is equal to 1 × 180 / 100 or 9 / 5 Fahrenheit divisions. Thus, 10 Celsius divisions will be = 10 × 9 / 5 Fahrenheit divisions.
So a reading of 10^{0} C in the Celsius thermometer will be shown as 10 × 9 / 5 + 32 in the Fahrenheit thermometer. Now instead of 10^{0} C in our example, we can generalize the relation and we get the formula
F = C × 9 / 5 +32
To convert the Celsius reading into the Kelvin scale, we add 273. Thus 20^{0} = (20 + 273) K or 293 K. Conversely, 423 K = (423 – 273 ) ^{0} or 150 ^{0} C. Here we subtract 273 from the Kelvin reading. Why we do so will be clear after we learn gas laws.
This is easy to understand for 100 Celsius divisions occupy the whole space that is occupied by 180 Fahrenheit divisions. So 1^{0} C division is equal to 1 × 180 / 100 or 9 / 5 Fahrenheit divisions. Thus, 10 Celsius divisions will be = 10 × 9 / 5 Fahrenheit divisions. So a reading of 10^{0} C in the Celsius thermometer will be shown as 10 × 9 / 5 + 32 in the Fahrenheit thermometer. Now instead of 10^{0} C in our example, we can generalize the relation and we get the formula
F = C × 9 / 5 +32
To convert the Celsius reading into the Kelvin scale, we add 273. Thus 20^{0} = (20 + 273) K or 293 K. Conversely, 423 K = (423 – 273 ) ^{0} or 150 ^{0} C. Here we subtract 273 from the Kelvin reading. Why we do so will be clear after we learn gas laws.
Different types of thermometers: The thermometer used by doctors is called the clinical thermometer. It is graduated in Fahrenheit degrees, the range being only between 95^{0} F and 110 ^{0} F. This limited range is due to the fact that our body temperature cannot be lower than 95 ^{0} or higher than 110 ^{0} F. In fact our normal temperature is 98.4 ^{0} F. To help read the temperature after taking the instrument from the mouth of the patient, a constriction is provided which prevents the mercury level from going down.
We can however bring it down by giving the thermometer a gentle jerk when it will be ready for use with another patient. Another type of thermometer is the maximum and minimum thermometer which can help us to record the maximum and minimum temperature reached during a certain interval of time, say during the day or night. The description and working of this thermometer can be found in any Physics text book.
The above thermometers depend for their working on the expansion and contraction of mercury on heating. The expansion of metals on heating can also be used for the measurement of temperature Gas thermometers also work on similar principle. The increase in resistance of metals on heating can also provide a basis.
Expansion of Solids: (i) While mercury which is a liquid can expand in volume only, solids can expand in length, area and volume. Expansion in length is called linear expansion, area expansion is known as superficial expansion and volume expansion is cubic expansion.
(ii) While the fact of expansion due to heat is important, scientists and engineers are more interested in the quantitative aspect of it, i.e. how much it expands. This leads us to the concept of expansivity or coefficient of expansion.
Coefficient of linear expansion of a solid i.e. the linear expansivity of the solid.
This is the ratio of the increase in length of a rod to it original length per ^{0} C rise of temperature. It is denoted by the letter alpha ( α ).
Thus we have
α = ( l_{2} – l_{1} ) / l_{1} (t_{2} – t_{1} )………………(1)
Where l_{1} is the length at t_{1} ^{0} C, l _{2} is the length at t_{2} ^{0} C. α is thus a ratio. Its value for copper is 17 × 10^{-6} per ^{0} C. Since it is a ratio, the units of length chosen in the (C.G.S. or S.I units) will not alter its value.
By using the formula () 1 above, we can find any unknown quantity if the others are known. However, formula (2) below which can be deduced from (1) may sometimes be found more useful.
α = ( l_{2} – l_{1} ) / l_{1} (t_{2} – t_{1} )
Cross multiplying
l_{2} – l_{1} = α l_{1} (t_{2} – t_{1})
Therefore, l_{2} = l_{1} + α l_{1} (t_{2} – t_{1})
= l_{1} [ 1 + α (t_{2} - t_{1}) ]
i.e. l_{2} = l_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………(2)
(iv) Similarly, the coefficients of superficial ( area ) expansion (β) and cubical (volume) expansion (γ) can be written as
A _{2} A_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………….(3)
V _{2} V_{1} [ 1 + α (t_{2} - t_{1}) ]…………………………………….(4)
It can be shown that β = 2α and γ = 3α
Thus α for copper is 0.000017 per ^{0} C. Its β and γ values are
2 × 0.000017 per ^{0} C and 3 × 0.000017 per ^{0} C.
To prove β = 2α
We take a square sheet of metal of side unit length. Its area = 1 sq. unit. Suppose it is heated by 1 ^{0} C, each side will now expand and become (1 + α) where α is the coefficient of linear expansion of the metal. The new area is (1 + α) ^{2} i.e. 1 + 2α + α^{2}. [α being very small can be neglected]. So the new area becomes 1 + 2α.
Therefore, increase in area = 2α.
And this increase being on unit area for 1 ^{0} C rise of temperature, it is β by definition.
Therefore, we have β = 2 α
To prove γ = 3 α
Let us consider a cube of a metal of side 1 unit of length. Its volume = 1 unit of volume. If it is heated through 1 ^{0} C, each side will expand to (1 + α) and so the new volume will be (1 + α)^{3} = 1 + 3 α + 3 α^{2} + α^{3} = 1 + 3 α, as 3 α^{2} and α^{3} being very small can be neglected. So the increase in volume = 3 α. As this increase is on 1 unit of volume for 1 ^{0} C rise of temperature, it is γ. So we have γ = 3 α.
- Posted by Max in Cause of Motion |
- December 12th, 2009 |
What is the cause for a thing to move or a moving thing to stop? It was Sir Isaac Newton who found the answers to this and related problems through his three famous laws of motion that are the basis for Newtonian or classical Physics.
Newton’s Three Laws of Motion
First Law: All bodies continue to be in a state of rest or of uniform motion in a straight line unless compelled by an external force to act otherwise.
Second Law: Rate of change of momentum is proportional to the impressed force and takes place in the direction in which the force acts.
Third Law: For every action, there is an equal and opposite reaction.
The First Law Explained
(i) Motion is caused or stopped by applying an external force. Without this external force, a body at rest or in motion will remain so for ever. It is this force that brings about the change. This concept thus gives us a definition of force. Force is that, which makes a body move or a moving body stop.
Force, thus tells about the nature of the body – its property by which the body tends to retain its status quo and does not like to be disturbed. This property is known as inertia.
(ii) Examples of the 1st Law: When you travel in a motor car and the latter suddenly stops, you are thrown forward. This is because when the car was in motion, you had the same speed as that of the car, but when it stops moving, you still retain your speed as that of the speeding car, and due to inertia, you still retain that speed for some more time even after it stops. So the position is, although the car has stopped, you still tend to move forward.
A person jumping from a running train is thrown forward and gts badly hurt for a similar reason.
The Second Law of Newton Explained
To understand this, we have first to know the meaning of momentum.
(i) Momentum of a moving body is the quantity of motion possessed by it and is equal to its mass multiplied by its velocity. Thus, we have M = mv, where M is the momentum, m its mass and v is velocity. All moving bodies possess momentum, the amount possessed by them being governed by the above equation.
(ii) The velocity of a body moving with acceleration changes with time and therefore its momentum too changes. Thus if m is the mass of body, u and v the initial and final velocities and t is the time, then mv – mu is the change of momentum in t sec and (mv – mu)/t or m(v – u)/t the rate of change of momentum. Now th 2nd law states that this is α F [the impressed force], i.e.,
m(v – u)/t α F
Things Deductible From The Second Law
(a) (v – u)/t = a (acceleration)
(b) So the above relation becomes
ma α F
or F α ma
which becomes F = ma, if the units are properly chosen.
The above law tells us
(i) Force acting on a mass produces acceleration on it.
(ii) The acceleration produced is proportional to the force.
Thus the second law is also referred to as the law of acceleration.
Definition of force and unit of force:
From the relation F = ma, if m = 1, a = 1, then F is also one. So we can define
(i) Force as that which acting on a mass produces acceleration on it in the direction of the force.
(ii) Unit of force is that force which acting on unit mass produces unit acceleration on it in the direction of the force. I S. I. system, unit of force is newton and in C.G>S. system, it is the dyne.
1 newton (N) = 10^{5} dynes
(iii) Newton: Newton (N) is the S.I. unit of force and is that force which acting on a mass of 1 kg produces on it an acceleration of 1 m/s^{2}.
(iv) Dyne is the C.G.S. unit of force which acting on a mass of 1 g produces on it an acceleration of 1 cm/s^{2}
Force is a vector quantity
Like velocity and acceleration, force is a vector quantity. Thus a force of 4 N can be represented by a straight line with an arrow.
——————|——————|——————|—————->
The figure shows a force of 4 N in the direction of the arrow shown.
Newton’s Third Law
There are numerous examples to support Newton’s third law of motion. Following are given few examples.
When we sit on a chair, our weight (force) presses on the seat. The seat in turn simultaneously presses on us. Out pressing on the chain is the action here and the pressing of the seat on us is the reaction.
As a rifle is fired, the bullet moves due to the forward force while the rifle recoils due to an equal backward force.
When we kick a ball, the ball moves forward due to the forward force of our foot, while our foot receives a pressure due to backward force from the ball.
An apple falls on the ground due to the gravitational force of the earth. The apple simultaneously attracts the earth with an equal force. However, the earth does not go up. This is because the force between the earth and apple is very small. This force is enough to pull the apple to the earth but is not enough to pull the earth towards the apple.
Conservation of Momentum
When a moving body A collides with another body B, A loses some of its momentum which the other gains. The gain is equal to the loss; so the total momentum remains unchanged. This principle is known as the law of conservation of momentum. It is understood that no external force comes into play.
If a body A having mass m_{1} and velocity u_{1} collides with another body B of mass m_{2} and velocity u_{2} and v_{1} and v_{2} are their velocities after collision, then
Sum of the initial momenta of the two bodies = m_{1} u_{1} + m_{2} u_{2}
Sum of the final momenta, i.e., after collision = m_{1} v_{1} + m_{2} v_{2}
It is found that these two values are equal, i.e.
m_{1} u_{1} + m_{2} u_{2} = m_{1} u_{1} + m_{2} u_{2}
Momentum is a vector quantity. It has both magnitude and direction.
Impulse
Impulse is a force applied for a short time and is represented by F × t, where F is the force and t is the time for which it acts.
For example, a blow or kick is an impulse. The effect of the impulse on the body is that the latter gains in momentum equal to it. Thus if m × v is the gain of momentum, then we have
F × t = mv
Impulse is measured in newton-second (N-s) and momentum in kg-m/s.
Mass And Weight
Any mass possesses weight when it is subject to the influence of gravity. This weight is a force and is equal to m × g. i.e. mass multiplied by acceleration. Here, the acceleration is due to gravity. If mass is 5 kg, then at a place where g = 9.8 m / s^{2}, its weight will be 5 × 9.8 newton i.e. 49 N. Conversely, if its weight is 19.6 N, its mass is 19.6/9.8 or 2 kg. Thus mass is matter and weight is a force.
Newton’s Law Of Universal Gravitation
It is known that earth attracts an apple on a tree and the apple too attracts the earth. Newton discovered that this mutual attraction applied not only to earth and apple, but to any two bodies including the heavenly bodies. He stated the law relating to this as follows:
The Law of universal gravitation: Everybody attracts every other body with a force which is directly proportional to the product of the masses and inversely proportional to the square of the distance between them, i.e., F α m_{1}m_{2}/r^{2} or F = G m_{1}m_{2}/r^{2}, where G is the constant of proportionality or the gravitational constant.
As the above law is universal, it is also known as the Law of Universal gravitational and the constant G as universal gravitational constant.
When F is in newton, m_{1} and m_{2} in kilograms and r in meters, G has the value of 6.67 × 10^{-11} SI units i.e. 6.67 × 10^{-11} N.m^{2} / kg^{2}
It is easy to see that the above value of G is very, very small.
Let two persons are talking to each other across a table, and each has weights of 50 kg and are separated by 1 meter, then the force of attraction between them will be
F = G × (m_{1}m_{2}/r^{2})
= ( 6.67 × 10^{-1} × 50 × 50 ) / 1^{2} = 1.67 × 10^{-7} N, which will be many times smaller than the force exerted by a small piece of paper on the pan of a balance. But of the two bodies, if one is the earth which has a large mass and other is you standing on the ground then the mutual force of attraction will not be negligible.
To Determine The Mass Of A Body
(a) Using Newton’s 2nd Law
Suppose you want to find the mass m_{1} of body A. Take another body B whose mass m_{2} is known.
We subject both m_{1} and m_{2} to the same force F and note the acceleration on each, say a_{1} and a_{2}. Then we have
F = m_{1} a_{1} = m_{2} a_{2}
Or m_{1}/ m_{2} = a_{2}/ a_{1}
Say a_{2}/ a_{2} = 3
Then we have m_{1} / m_{2} = 3 ; or m_{1} = 3 m_{2}
We already know m_{2}
Therefore, m_{1} is double that value of m_{2}
So in this method, the ratio of the accelerations helps you to find the unknown mass.
(b) If m_{1} and m_{2} are the two masses, the forces of attraction by the earth on them are given by
F_{1} = G M_{e} m_{1} / r^{2} and
F_{2} = G M_{e} m_{2} / r^{2}
where M_{e} is the mass of earth.
If F_{1} = F_{2}, then we have
G M_{e} m_{1} / r^{2} = M_{e} m_{2} / r^{2}
Or m_{1} = m_{2}
So the problem is to make earth’s attraction on both of them to be the same.
This is secured by a physical balance.
When we secure the balance and the pointer is in the middle of the scale, earth is attracting both the pans and contents equally. In this condition, the body whose mass m_{1} is to be found exactly balances the other body whose mass m_{2} is already known.
Weight of a body on earth and on the moon compared:
Suppose the mass of the body is m. It will be the same both on earth and on the moon. Its weight on the earth F_{e} = G × M_{e} × m/ r_{e}^{2} by Newton’s universal law of gravitation. F_{e} is the weight on earth, M_{e} mass of earth, m mass of body and r_{e} radius of earth.
Its weight on the moon F_{m} = G × M_{m} × m / r_{m}^{2}
Dividing (2) by (1), we get F_{m} / F_{e} = M_{m} / M_{e} × r_{e}^{2} / r_{m}^{2}
The mass of earth is about 100 times greater than that of the moon and its radius 4 times.
Hence, F_{m} / F_{e} = 1/100 ×4^{2}/1 = 1/6 roughly.
Therefore, weight of body on the moon = F_{e} / 6 i.e. the weight of the body on the moon is only 1/6 of its weight on earth.
Change in weight due to acceleration and deceleration
When we go up in a lift which is accelerating upwards, we feel heavier, i.e. there is an increase in our weight and when the lift is accelerating downwards, we feel lighter. Why?
In an up-going accelerating lift, the man experiences the same acceleration as the lift say ‘a’ and hence a force ma in the direction of motion where m is the mass of the man. This force by Newton’s third law of motion exerts and equal downward force on the floor of the lift.
Then there is also the force mg – the normal weight of the man’s body pressing on the floor of the lift where g is the acceleration due to gravity. Thus the total or resultant force that the man’s body exerts on the floor is ma + mg or m (g + a) which is more than his normal weight mg. So he feels heavier.
When the lift accelerates downward with the same acceleration ‘a’, part of his normal weight mg is utilized in giving him this acceleration and only the balance is left as his resultant weight which will be mg – ma or m (g – a) and this will be zero when g = a as in free fall.
Weightlessness: Weightlessness is the common experience of the astronaut during most of their journey in space. They feels it during orbital motion, during free fall or where their gravitational weight is balanced or neutralized by an opposing force.
In orbital flight, the weight of the astronaut is just sufficient to provide the centripetal force to keep him in orbit leaving no net force to provide his or her weight. So he feels weightless. In free fall, the acceleration of the astronaut and the capsule is the same namely ‘g’ and in this condition his weight will be zero.
To find an expression for acceleration due to gravity (g) at a given place using Newton’s law of universal gravitation
Let us consider a body of mass ‘m’ on earth.
The force of attraction F on it by the earth is given by
F = G × Mm / r ^{2}
M is the mass of earth, m mass of body and r radius of earth. This force acting on the mass will produce an acceleration ‘g’ on the mass by Newton’s 2nd law.
G = F / M = G. (Mm / r^{2}) / m
= G.M / r^{2}
This acceleration is known as acceleration due to gravity and depends on the mass M of the earth, the distance of the body from the center of the earth (if the body is a few meters high above the earth, this distance can be ignored and r can be taken as the radius of the earth).
Note the value of ‘g’ does not depend on the mass of the body; the formula clearly shows that, for it does not contain ‘m’. Galileo proved this fact by his famous experiment from the leaning tower of Pisa.
- Posted by Max in Linear Motion |
- December 11th, 2009 |
Linear Motion Let us begin our discussion on linear motion with few definitions.
Velocity Velocity of a moving body is the rate of change of position of the body in a straight line or the rate of displacement in a given direction. Velocity is expressed as meters per second (m/s). It is a vector quantity.
Vectors: Physical quantities that have both magnitude and direction are known as Vectors.
Vectors are represented by a straight line with an arrow. Examples of vectors are Displacement, Velocity, Acceleration and Force.
Vector quantities are represented by an arrow over them.
A ————1————–1————→ B East
The above figure represents a velocity of 3 meters per second. The magnitude is 3 units of length per second in the direction AB, i.e., East.
Scalars: Quantities that have a magnitude but no direction are known as Scalars. Examples of scalars are Speed, Mass, Volume, Density etc. Suppose a person starts from A and moves along a curved path ACB of distance S meters and reaches B in‘t’ seconds, then the average speed of the person is S/t m/s.
However, the displacement is only AB, which is the straight line distance from A to B that has taken place in ‘t’ seconds. Hence, the velocity of the person is only equal to AB/t m/s, that is very much different from the Speed S/t .
Uniform and variable velocity: Velocity of a moving body is said to be uniform if the body moves equal distances in equal interval of time or if velocity is same at all points of the motion.
On the other hand, if the velocity changes with time, then it is variable velocity.
Acceleration Acceleration is defined as the rate of change of velocity, which means how much the velocity changes per second. Acceleration is said to be uniform if the change of velocity is same throughout. If it is not, then acceleration is variable.
Acceleration is measured in meter per second per second (m/s/s) or in a shorter form m/s^{2}. If the velocity increases every second, then the acceleration is positive. If it decreases every second, then it is negative acceleration or retardation.
Motion in a straight line is known as Linear Motion
Equation 1: This equation provides the relation between initial velocity, final velocity, the time taken to reach it, and the acceleration.
Let the initial velocity be u, the final velocity v, time taken t, and acceleration ‘a’, then the change of velocity = v – u
The time taken for this change = t
Therefore, the rate of change of velocity, i.e., change of velocity per sec
or acceleration a = (v – u)/t i.e., v = u + at
Note: If there is retardation, we use ‘–a’ in place of ‘a’.
Equation 2: This equation provides the relation between distance travelled, initial velocity, acceleration and time.
Let the initial velocity be u
Let the final velocity be v
Let the time taken be t
And, let the distance travelled be S
Now S = average velocity × time = [(u + v)/2]/t
But v = u + at {from equation 1}
Substituting for v, we get S = [(u + u + at)/2]×t
= ut + at^{2}/2
Hence, S = ut + at^{2}/2
The same formula can be used for retardation also, but we need to put ‘-a’ in place of ‘a’.
Equation 3: This equation provides the relation between initial velocity, final velocity, acceleration and distance. There is no time factor in it.
v = u + at
Squaring, we get v^{2} = (u + at) ^{2}
= u^{2} + 2uat +a^{2} t^{2}
= u^{2} + 2a (ut + a t^{2}/2)
= u^{2} + 2aS
Hence, v^{2} = u^{2} + 2aS In case of retardation, ‘a’ becomes ‘-a’
Note: To find the distance travelled in a given time, we make use the formula
S = ut + at^{2}/2.
If it is required to find the distance travelled in the third second, then using the formula, we find the distance travelled in 3 sec and in 2 sec and then subtract the value of the second from the first.
However, we can find a single formula to find the distance travelled in the‘t’ th second as given below:
To find the distance travelled in the‘t’ th second
We have to first find the distance in t sec and in (t-1) sec, subtract and simplify. We will get the required formula.
S_{t} = ut + at^{2}/2, where S_{t} is the distance travelled in t sec.
S_{t-1} = u(t-1) + a(t-1)^{2}/2 where S_{t-1} is the distance travelled in t-1 sec.
S_{t} – S_{t-1} = ut + at^{2}/2 – u(t-1) – a(t-1)^{2}/2
i.e., S_{’t’th}= ut –u(t-1) + a[t^{2} - (t-1)^{2}]/2
= u + a(t^{2} – t^{2}+ 2t – 1)/2
= u + a(2t – 1)/2
= u + a(t – 1/2)
By using the above formula, we can calculate the distance traveled in any particular second.
Thus, when u = 2m/s and a = 0.5 m/s^{2}, the distance travelled in the 3rd sec is S_{3rd} = 2 + 0.5 (3 – 1/2) = 3.25 m
All freely falling bodies are subjected to a uniform acceleration. Its values at the equator and the poles are different. Its value is usually taken to be 9.8 m / s^{2} which is denoted by ‘g’. The three equations of linear motion apply to falling bodies as well. To apply, we have to replace ‘g’ for ‘a’.
Thus the three equations of motion in case of falling bodies are
V = u + gt
S = ut + gt^{2}/2
v^{2} = u^{2} + 2gS
If a body is projected upwards, we have to use ‘-g’ for ‘g’.
Example: A body starting from rest and executing an accelerated motion covers a distance of 9 cm in 6 seconds. Calculate (a) the acceleration (b) the final velocity.
(a) S = ut + at^{2}/2
Implies, 9 = (0×6) + (a×6^{2})/2
Therefore, a = 0.5 m/s^{2}
(b) V = u + at = 0 + (0.5×6) = 3 m/s
A bullet is fired vertically with an initial velocity of 29.4 m/s (a) How high will it reach? (b
) What is the time taken to reach that height?
u = 29.4 m/s; v = 0
g = -9.8 m/s^{2}
S = ?
t = ?
(a) v^{2} = u^{2} + 2gS
Implies, 0 = 29.4^{2} + 2×(-9.8)×S
Implies, S = 2×9.8 = (29.4)^{2}
Implies, S = (29.4)^{2}/(2×9.8) = 44.1 m
(b) v = u + gt
0 = 29.4 + (-9.8) t
Therefore, t = 3 sec
- Posted by Bertha in Grammar |
- December 3rd, 2009 |
Prepositions are words placed before a noun or a pronoun to show what one thing or one person has to do with another thing or person; like: -
i) My hand is on the table.
In the above sentence, if you omit the word on, then the sentence makes no sense. You can place your hand on the table, or under the table, or above the table. Unless you add some Preposition to the sentence, the relation between the table and the hand is not clear.
ii) You are in a good mood today.
In the sentence, the word in is placed before the noun ‘mood’ (or ‘a good mood’) and shows what you have to do with a good mood. Therefore, in is a Preposition.
iii) You arrived here before me.
In this sentence, the word before is placed before the pronoun ‘me’ and it shows what your arrival has to do with ‘me’. It shows you arrived sooner than I did. Therefore, before is a Preposition.
Prepositions are never added to any Part of Speech other than a noun or a pronoun or their equivalent.
Prepositions often have same form as the Adverbs. So, how can you differentiate? Here is a simple rule that can help you out to solve this problem and that is;
Adverbs are never added to a noun or a pronoun.
Following this rule you can tell whether a word is an Adverb or a Preposition. Some examples are given below;
Prepositions |
Adverbs |
I walked about the field. |
I walked about. |
The sky is above the earth |
The above-mentioned name. |
The man lives down the lane. |
Sit down there. |
Let me walk along the road. |
Go along slowly. |
The pen is inside the pencil box. |
She sat inside. |
By whom was the book written? |
The cow was grazing by. |
Fish swim in the water. |
Mosquitoes fly in and out. |
He slept within the room. |
The room was never clean within. |
His house is near yours. |
They are standing near. |
Since that year she has been ill. |
She passed away three years since. |
She went after a few days. |
She went a few days after |
The noun or the pronoun, which is placed after a Preposition, is called an Object.
- Sometimes two Prepositions are used together, but both having the same object; as,
The rat crept in between the cardboards.
The rat appeared from between the cardboards.
The man stood out from among all.
She came from within her room.
- Sometimes, a Preposition takes form of a phrase, instead of a single word. However, a Prepositional Phrase always ends in a Simple Preposition.
In front of; because of; for the sake of; in the event of, with regard to; on behalf of; with reference to; in the place of; with a view to; on account of; by means of; in opposition to;, because of; in lieu of;, instead of.
- Sometimes, the object to the Preposition is an adverb used as a noun and sometimes is a sentence.
Till then; from here; from now; before now etc. (Adverbs)
She told everybody of what she had done. (Sentence)
- Posted by Paula in Reading and writing skills |
- December 3rd, 2009 |
Are you cramming your head with facts the evening before that big exam? You will succeed in reading only some of that book-and learn less! Wonder why? Your concentration is least when you are trying to cram your brain with data! It has a tendency to wander when you pressurize it! So what do you do? How do you get the best out of reading, and invest the least amount of time while doing so? You ‘speed read’!
What’s Speed Reading?
Well, it is a speedy way of reading text and getting the most out of it. More and more researchers are advocating these techniques to get the most out of reading! Wouldn’t you love to spend less time at your homework-but get the most out of it? Well, the only way out is to read right-and here’s how you go about it!
How Should I Read and Get the Best Out Of It in A Short Time?
- Peace and quiet-The most important thing you need to remember about this is that you should find out the best place for it. For instance, look for peace and quiet-that goes without saying! You can NEVER learn anything without this! Relax-I include your favorite music in the ‘peace and quiet’ I’m talking about!
Having said that, I seriously doubt your powers to concentrate when you are listening to reggae or hard rock at the max! I have nothing against it- but are you sure it helps your concentration?
- Relax as you read-like we told you before, your brain feels pressurized when you task it too much. Don’t concentrate on cramming facts. Your mind will wander and you will remember precious little!
If you are having problems concentrating- you’re just normal. Why not take up a few minutes of meditation every day? You will find it working miracles! Just about 5 minutes in a day is more than enough at the outset. Surely you can afford that time?
- Plan ahead- you need to be clear about your expectations from the text. For instance, if you are writing an essay and are looking for matter, jot down key words and ideas as you read. You will find it easier to assimilate the ideas later! If you are preparing for a test, this will help you cut down the hours you spend at your table.
- Read in groups- if you read through each word in the text, you will waste time and concentrate less. Instead, try and read the words in groups.
For instance, if you read ‘if you want to concentrate more, read slow’, in individual words, you will just waste time. Instead, why not try and read them like this- ‘if you concentrate more, read slow’! Try it out-it’s fun!
Remember, you need not spend hours at your desk- even before that exam! Just read right and you will get away with the least amount of time-and we all know what you can do with that time! But then, my mind is wandering……
Don’t just do it fast….do it right! Read this quick step
http://tutorteddy.com/wordpress/tips-on-reading-tips-on-writing/4-steps-to-reading-right/