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## Cause of Motion

1. Posted by Max in Cause of Motion |
2. 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) = 105 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/s2.

(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/s2
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 m1 and velocity u1 collides with another body B of mass m2 and velocity u2 and v1 and v2 are their velocities after collision, then
Sum of the initial momenta of the two bodies = m1 u1 + m2 u2
Sum of the final momenta, i.e., after collision = m1 v1 + m2 v2
It is found that these two values are equal, i.e.
m1 u1 + m2 u2 = m1 u1 + m2 u2

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 / s2, 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 m1m2/r2 or F = G m1m2/r2, 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, m1 and m2 in kilograms and r in meters, G has the value of 6.67 × 10-11 SI units i.e. 6.67 × 10-11 N.m2 / kg2
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 × (m1m2/r2)
= ( 6.67 × 10-1 × 50 × 50 ) / 12 = 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 m1 of body A. Take another body B whose mass m2 is known.

We subject both m1 and m2 to the same force F and note the acceleration on each, say a1 and a2. Then we have

F = m1 a1 = m2 a2
Or m1/ m2 = a2/ a1

Say a2/ a2 = 3
Then we have m1 / m2 = 3 ; or m1 = 3 m2

Therefore, m1 is double that value of m2
So in this method, the ratio of the accelerations helps you to find the unknown mass.

(b) If m1 and m2 are the two masses, the forces of attraction by the earth on them are given by

F1 = G Me m1 / r2 and

F2 = G Me m2 / r2
where Me is the mass of earth.

If F1 = F2, then we have

G Me m1 / r2 = Me m2 / r2

Or m1 = m2

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 m1 is to be found exactly balances the other body whose mass m2 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 Fe = G × Me × m/ re2 by Newton’s universal law of gravitation. Fe is the weight on earth, Me mass of earth, m mass of body and re radius of earth.

Its weight on the moon Fm = G × Mm × m / rm2
Dividing (2) by (1), we get Fm / Fe = Mm / Me × re2 / rm2

The mass of earth is about 100 times greater than that of the moon and its radius 4 times.

Hence, Fm / Fe = 1/100 ×42/1 = 1/6 roughly.

Therefore, weight of body on the moon = Fe / 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 / r2) / m

= G.M / r2

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.

## The Three Equations Of Linear Motion

1. Posted by Max in Linear Motion |
2. 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/s2. 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 + at2/2

Hence, S = ut + at2/2

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 v2 = (u + at) 2

= u2 + 2uat +a2 t2

= u2 + 2a (ut + a t2/2)

= u2 + 2aS

Hence, v2 = u2 + 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 + at2/2.

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.

St = ut + at2/2, where St is the distance travelled in t sec.

St-1 = u(t-1) + a(t-1)2/2 where St-1 is the distance travelled in t-1 sec.

St – St-1 = ut + at2/2 – u(t-1) – a(t-1)2/2

i.e., S’t’th= ut –u(t-1) + a[t2 - (t-1)2]/2

= u + a(t2 – t2+ 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/s2, the distance travelled in the 3rd sec is S3rd = 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 / s2 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 + gt2/2

v2 = u2 + 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 + at2/2

Implies, 9 = (0×6) + (a×62)/2

Therefore, a = 0.5 m/s2

(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/s2

S = ?

t = ?

(a) v2 = u2 + 2gS

Implies, 0 = 29.42 + 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

## Mechanics

1. Posted by Steven in Mechanics |
2. November 12th, 2009 |

MOTION

Science  of   motion  is  a very  important  aspect  of  the  branch  of  physics. It is a common rule of the physical world. But  before  having  a thorough  discussion on this  topic  one  must  have  a  common  concept  in  his /her  mind  that, practically every motion is in a relative  state. The motion  we  are  talking  about is  depending on the  place  occurred  by the object & the  observer at that  particular time.

Comparative discussion between Rest & Motion:

When do we call an object that if it is in rest or motion? Again,  I have to tell you  it is dependency on the  way  or view  we  are  considering from. Suppose there is flower vase on the table. If after few seconds we are seeing that particular object in the same place we can tell that the flower vase is in rest.

Actually we are considering the object with the relative frame of the room. So mathematically, the co-ordinates are not changing with time. So we are seeing that particular vase in the same place. But if we will consider that object from the moon or from any place outside the earth we will notice that the object is changing place with time, as earth is moving!

Hence  we  can  infer  from  the  above  mentioned  fact  that ,there  is  no  absolute  rest / absolute  motion. This is what depending on the frame of the viewer & object.

If relatively the object and the observer is not changing place we can tell, the object is in rest or if it is changing the object is in motion.

Another  instance  to describe  the  science  of  motion and  rest  is  like  that ,when  two  friends  are   going  by  a car  the substances  in  the  car  are  not  moving  with  respect  to  the  boys. But  the  substances  are  changing  with  the  frame  of  places  where  the  car  is    moving. So with  respect  to  the  boys we  can  say  the  objects are  in rest but  in motion  with  the  frame of places.

So to  locate  a  particular  object  we  need  some  co-ordinates(x,y,z) that  is called frame  of  reference. We cannot tell whether an object is in motion unless we have a frame of reference. A reference frame is another substance with respect to which we compare another object’s position.

The branch of Physics, dealing with the nature of moving objects is known as mechanics. Mechanics is divided into two parts namely Kinematics and Dynamics. Kinematics deals with the study of motion without taking into consideration the cause of motion, while Dynamics is dealing with the cause of motion, that is force.

Here we are concerned about the kinematics only.

Distance and Displacement

Suppose a particular object is traversing from point A at time t1 to point B at time t2 via the path A-C-D-B. So the  time  taken  by  the  object  to reach  from  A to B is (t2-t1).

Here we can tell that the distance traversed will be A-C-D-B. But the displacement will be  A-B.

The  length  of  the path  A-C-D-B is  called  the distance during  the  time  interval t1 and  t2. It’s a scalar quantity as  it  considers the  magnitude only.

The magnitude of the displacement is the length of the straight line joining the initial and final position i.e. A – B. It’s a vector quantity as it is considering both the magnitude and direction. It follows triangle rule for vector addition too.

Eva is taking a path from West Gate to East Gate i.e. of 100 m long. Alan is also going from West gate to East gate but he is taking a different way i.e. elongated too. Here from the above diagram we can see the fact. Here the displacement is same for both of them. But the distance traveled by Alan is much more than Eva.