## December 28, 2015

### MECHANICS

1. Introduction – The origin of our understanding of motion or mechanics

Motion in Physics means movement of bodies. Here in this section we will look at motion or movement of solid bodies.

When does a body move?

Let us start with our own observations of things, when they move.

The first kind of motion and the one that strikes us first is when one body hits another or there is a push or a pull given to bodies.

Kicking a football, opening a door, striking a carrom coin, throwing a ball, etc. etc. These can be called contact forces in gross terms became one body when striking another through actual contact does move the other body. Of course the body to move must be light enough and / or the contact force applied must be strong enough.

For example a large rock cannot be moved by a man but can be moved by a crane. But a small rock can be moved by a man. These are the things that we observe.

We also observe things falling from a height when left free. Of course birds and airplanes seem to go contrary to this but that is a phenomenon involving air and we won’t go into that now as in this section we are dealing with specifically motion involving solid bodies.

And then we see the heavenly bodies moving – the Sun, rising in the East, moving across the sky and setting in the West, the moon also moves across the sky but most people do not observe the moon moving as we sleep during that time! But we do see the moon at different places in the sky at different times when we do happen to observe it during late nights and very early in the morning.

We also see the countless stars and among these are the planets that also look like stars but are planets really. Usually people cannot distinguish between planets and stars but the interesting thing is that to us, the planets move during the year ( if you locate it and observe it the whole year) but the stars don’t seem to move.

Then we also see certain objects like magnets that move other objects like iron when brought near it. Travellers know another phenomenon that if you hang a magnet or a compass needle is left free, it moves and points in a specific (N-S) direction.

These are our observations and these same observations led the ancients to wonder and ask questions about all these motions.

Since we happen to see these things from childhood, we take them for granted we tend to feel that it is like that because it is like that!

Yet, there is a wonder to the observations just made. Setting aside contact forces for the moment, let us consider bodies falling when left free from a height.

Why should bodies fall at all? After all, nobody is pushing it. See the phenomenon as if you are seeing it for the first time. It is a wonder. Invariably, always, without fail, bodies fall on earth, if left free from a height.

The apparent motion of the Sun is even move wondrous and raises many questions.

What is the Sun? Why is there night and day? How far is it? Is the earth moving or the Sun moving? Why does it seem to move across the sky? Why is its apparent motion so regular?

Why do the moon and the planets too move as seen by us?

These questions were asked by the ancients. But some, the real seekers of knowledge, the more passionate of them did not only ask questions, they located the positions with the passage of time. It is equal to measuring the motion.

They found no irregularities with respect to the Sun and the moon. The Sun and the Moon went across the sky rising from the east and setting in the west as seen by us from Earth.

But when it came to the planets, they found a strange behavior of planets.

Let us take Mars. When they observed Mars the whole year, they found Mars to go in one direction till June and then it went backward for 3 months and then went forward again. This is how Mars looks to us.

The same is true of all the planets. This is called retrograde motion.

Now the central question is the following. Is the earth at the centre and all planets, Sun, Moon are moving round it or Is the Sun the centre and the planets move around it?

If the earth is at the centre, the planets should move across the sky, why is there backward or retrograde motion of the planets?

The ancient thinkers especially Ptolemy gave a system of Earth as the center and gave complicated motions to the planets.

But in the 16th century, it was Copernicus who got the real idea.

He proposed that the Sun is at the centre and the planets are moving round the Sun. He did this by explaining backward or retrograde motion of the planets.

If the Sun is at the centre and let us take 2 planets Earth & Mars are going round the Sun, then how will Mars look to people on Earth? This was the question that Copernicus asked.

When earth and Mars are side by side (Point A) and if Earth goes faster than Mars, then how will Mars look to people on Earth?

It will seem to go backward!

When earth and Mars are as shown in and Mars is at point B i.e. at opposite ends, Mars would seem to go forward!

So at some period Mars goes forward and during some period, it goes backward. Putting the Sun at the centre and the planets going around it explains the backward motion of all Planets as seen from earth.

The planets are not going backward and forward. They are simply going around the sun. it seems to go like that only because earth also is moving!

Now this was a great turning point in understanding motion.

Once we understood that Sun is at the centre, the next step was to state exactly, with measurement, as to how planets go around the sun.

Kepler did that work and came up with 3 Kepler’s laws.

Another great thinker Galileo came next and he measured free fall of bodies.

His interest was not just observing free fall but to measure how much distance it travelled while falling as time passed (on successive seconds).

He got the value of the acceleration as 9.8m/s2.

Now let us see how the picture looked at that time concerning motion.

The sun was at the centre and all the planets moved around it. (including earth). Things fell on earth with an acceleration of 9.8m/s2. The planets moved according to Kepler’s laws.

But the questions still remained. Why did the planets move like that? Why did the acceleration on earth of freely falling bodies have that value?

It was then that Newton came on the scene. He thought intensely and deeply over the problem, One day, the legend says, he was walking in an apple orchard and he saw an apple falling and it suddenly occurred to him that the moon falling or being pulled to earth by earth is the same pull that earth puts on the apple!

He further reasoned that the force is due to the product of masses & inversely proportional to the distance between the masses.

i.e. the earth pulls on the apple and the moon also. The force in the apple is

directly proportional to mass of apple x mass of earth.

Inversely proportional to square of distance (i.e.radius of the earth)

The force on the moon is

directly proportional to mass of moon x mass of earth

& inversely proportional to square of distance (i.e. distance between earth and moon)

Newton could, with this insight derive Kepler’s laws and at one stroke Newton made earth & heavens one!

He put all motion in the form of 3 laws of motion & called the pull of gravity between masses the universal law of gravitation.

Let us learn these laws systematically and mathematically. But before we go into it we must understand certain concepts of motion in one dimension. That will make our understanding of Newton’s laws complete.

The next section is on motion in one dimension. After that we cover Newton’s laws of motion and Universal law of Gravitation.

2. Motion in one dimension

2.1. Distance and Displacement

Look at the figure.

Suresh begins, at point A, from his house and travels on the road and takes turns crossing many houses and reading point B, his friend, Ramdas’s house.

A bird from a terrace also moves in the air from the terrace of Suresh’s house and goes to Ramdas’s terrace in a straight line.

The total distance travelled by Suresh is is called Distance. It has no direction.

The straight line distance from the initial point A towards the final point B (of the bird) is called Displacement. It has direction also. The direction is from A to B.

The figures-3&4 below illustrate this point further.

Units of both distance and displacement are the same. It is the unit of Length.

10 mm = 1 cm 100 cm = 1 m 1000 m = 1 km

1 feet = 12 inches 3 feet = 1 yard

2.2. Speed :

We know what is speed when we look at two objects moving; we can tell which is going faster. But what is involved in speed? How can we measure speed exactly?

Suppose two men come to you and the first man says, “I have travelled 100 metres”. The second man says “I have travelled 100 metres”. Just with this information, will we know who travelled faster? What else is required to know the faster of the two men?

Yes, the men must specify the time also. So the first man says “I travelled 100 metres in 50 seconds”. The second man says “I travelled 100 metres in 20 seconds”.

First man : 100 metres in 50 seconds.

Second man : 100 metres in 20 seconds.

So, in one second,

First man travelled = 100/50 = 2 m/seconds

Second man travelled =100/20 = 5 m/seconds

So second man is faster. In general if someone or something travels ‘d’ metres in ‘t’ seconds then in 1 second, i.e.

Speed=distance/time

The unit is m/sec or feet/sec or time/hr etc.

2.3. Velocity

Velocity too is speed but velocity talks about the displacement (not distance) in one second.

so, Velocity=Displacement/Time

So velocity measures how much final (net) straight line distance the body travelled (from initial to final point) in one second i.e. displacement divided by time.

eg. if the displacement was 10 metres in 20 seconds

in one second displacement = 10/20=1/2m/s

In general Velocity=displacement/time

2.4. Uniform Speed

Uniform speed is simply constant speed i.e. the body is covering equal distances in equal intervals of time.

eg: A man is going on a scooter. His hand on the accelerator is steady. He is neither raising the accelerator not lowering it. His speedometer shows 30 km/hr (8.3 m/s)

Now what does this mean?

It means that at every second he travelled 8.3 metres only.

1 sec 1 sec 1 sec 1 sec 1 sec 1 sec

8.3 m 8.3 m 8.3 m 8.3 m 8.3 m 8.3 m

This is the meaning of covering equal distances in equal intervals of time.

2.5. Average Speed

Usually bodies do not go at uniform or constant speed, at least on bikes!

See the figure below:

A caterpillar moving

1 sec 1 sec 1 sec 1 sec 1 sec 1 sec 1 sec

1 c m 2 cm 0 cm 1 cm 1 cm

A caterpillar moved 1 cm in 1st second,

2 cm in 2nd second,

again 0 cm in 3rd second, (it stopped for a second)

1 cm in 4th second,

1 cm in 5th second,

3 cm in 6th second,

2 cm in 7th second.

Now what is it’s speed? We will have to give seven speeds here, isn’t it?

That is difficult. But I still want how ffast the caterpillar travelled.

So we can take how much total distance it travelled in the total time.

10 cm in 7 seconds

so speed = 10/7 cm/sec

In general average speed = Total distance travelled/total time taken

2.6. Uniform velocity, Average velocity

Uniform velocity means that equal displacements take equal ........ls of time.

To find the Average velocity we use the formula

Average velocity = Total displacement/total time taken

2.7. SIGN NOTATION

Displacement and velocity being a vectors have two signs (for straight line motion). If we take a right side motion as positive, then the left side motion is (-)

V-------------------------------------------V

If we take upside motion as (+)ve then down side motion is (-)ve.

|

Let us see why

A man moves ahead (for displacement in a straight line) 4 metres and then moves back in the same straight line 3 metres. See figure...

--------------------------------------------->C A--------------------------------->C

A-------------------------------------------->-------- --------------------------------------

B-------------------------------------4mts B-------------------------------

This displacement from A to C is +4 metres

This displacement from A t o B is -3 metres (opposite direction)

So total displacement is 4 - 3 = 1 m.

If we had not put the signs.

We would have got 4 + 3 = 7 x

It would have been a wrong answer! So clearly displacement has (+)ve and (-)ve signs.

Since velocity is only displacement (in one second) velocity too has (+)ve and (-)ve signs.

2.8. Acceleration - Non-Uniform and Uniform Acceleration

In real life, usually, a body changes it’s velocity. It may go faster, slower or come to a stop. This is clear enough. This is called acceleration.

But how do we measure this?

Usually, bodies do not go at a constant speed (as we saw in the caterpillar example). Suppose I am going on my bike. How do I go? I am at 20 km/hr now, I accelerate and reach 40 km/hr in one second. Then I stop at a red light. Then again I came to 20 in one second. I raise......

So here we have to give values and numbers at every moments. This is called non-uniform acceleration.

There can be also uniform acceleration.

Look at the following example

A body moves in the following way...

1sec 1sec 1sec 1sec 1sec 1sec 1sec 1sec Time

-----------------------------------------------------------------------

2m 4m 6m 8m 10m 12m 14m 16m Distance

What do you observe?

In 1st second it moves - 2 m

2nd second it moves - 4 m, an increase of 2m

3rd second it moves - 6 m, a further increase of 2m

4th second it moves - 8 m, an increase of 2m

10 m, an increase of 2m

12 m, an increase of 2m

14 m, an increase of 2m

16 m, an increase of 2m

With each second the distance moved is 2 m move per second.

The change in velocity per second is 2 m/s.

If a body starts with a velocity of 2 m/s and after 10 second reaches a velocity of 10 m/s. then, change in velocity is (10 - 2) m/s in 10 seconds = 8 m/s in 10 seconds.

Hence in one second, the change in velocity is 8/10 = 0.8 m/s/s

In general if a body starts with a velocity ‘U’ and reaches a velocity of ‘V’ in ‘t’ seconds. The change in velocity = V - U in ‘t’ seconds.

in one second,

change in velocity =V-U/T

a = V-U/t m/8/s or m/s

Note that the above is true in Uniform Acceleration only.

i.e. change in velocity is same through out the motion. Now, Average velocity is U+V/2

Displacement = Average Velocity x time

=(U+V/2)t

=(U+U+at/2)t

(S=Ut+1/2 at2)-2

V=U+at S=ut+1/2at2

t=(V-U/a)

t2 = U2=V2-2UV/a

S=U(V-U/a)+1/2a(V2+U2-2UV/a)

= 2UV-2U2+V2+U2-2UVa-2UV/2a

S=V2-U2/2a

V2 - U2 = 2as ——— 3

3. Newton’s Laws and universal law of gravitation

Newton was the central figure in the history of physics. He was the person who made Mechanics into a ‘whole’, comprehesive science. He gave the fundamental statement to it.

What is it that Newton said?

Lets begin with his famous 3 laws of motion

3.1. Newton’s First Law (statement)

“Every body remains at rest or of uniform motion in a straight line unless influenced by an external force”.

A body remains at rest if it is at rest. Ok that’s fine. That goes with common sense and is pretty plain and obvious.

But what about the 2nd part - (A body) remains in 1. Uniform motion, 2. In a straight line, 3.Unless disturbed by an external force.

Is this true?

On Earth, we see bodies stopping even when nothing is stopping it. A ball rolled on a floor does come to a stop after moving some distance by itself.

Actually on Earth, the bodies are not free. Either the surfaces or air opposes any motion. When a body rolls on the floor, it is not really free because the roughness of the floor is showing it down. It is opposing the motion. This opposing force is called Frictm.

But think of what would happen if there were no force opposing a moving body. A region like space where there are no bodies and no air would be a frictionless area. What if a body is given a small push and then left alone. You will actually see the body moving with the same speed and NEVER STOPPING, unless something stops it, NEVER GOING FASTER or SLOWER unless something makes it go faster or slower. If a push is given in the same direction of its motion it’ll go faster. If a push is given in the opposite direction, it will go slower. If it is given at an angle only then it’s direction will change.

A body remains at rest or

in uniform motion

in a straight line

unless it is influenced by some external force.

You understand? In the universe, as such, the first law is true.

A body by itself cannot change it’s state of rest and also of motion.

It’s a wondrous thing, isn’t it? Since childhood, we are used to thinking that a body in motion will come to a stop ultimately. But now, I hope, you understand that it comes to a stop on Earth only because of an opposing force called Friction.

That’s why if the Friction is less, say on a smooth floor, the body takes longer to come to a stop. Making Friction zero and leaving the moving body totally undisturbed will make the body continue to move forever and ever....

You see motion too is natural like rest.

A body at rest will remain at rest.

A body in motion will remain in motion!

This is the principle of satellities and the motion of planets and moon etc. Once a particular motion was given, there was nothing to stop it! The Earth once was given rotation around itself and revolution around Sun and it is doing the same thing and will do the same forever and ever and ever!

Give any motion to a body it will be in that motion forever in the universe. We don’t see it happening on Earth because of surface friction.

3.2. Newton’s Second Law

“A force acting on a body accelerates it; greater the force, greater the acceleration. Greater the mass of the object, the lesser the acceleration FOR THE SAME FORCE.”

Putting it another way

The acceleration of a body is directly proportional to the force and inversely proportional to mass.

(Directly proportional means if one is increased the other increases proportionately i.e. if one is doubled the second is doubled. If one is tripled, the second is tripled and so on.

Inversely proportional means if one is increased the other is decreased proportionately. If one is doubled, the other is halved, when one is tripled, the other becomes one third)

Let us try to understand the 2nd Law

A marble is at rest on a floor.

If I push it, it moves.

A marble is moving slowly on a floor.

If I push it in the direction of motion of the body, it goes faster.

If I push it in the opposing direction, it goes slower.

If I push it at an angle to it’s motion it changes direction.

So 4 things can happen to a body when a force is applied to it.

(1) It already on motion it can go from rest to motion.

(2) go faster

(3) go slower

(4) change direction

depending on the direction of the force.

Same direction to the direction of motion - faster

Opposite direction to the direction of motion - slower

at an angle to the direction of motion - changes direction.

Going faster or slower is called acceleration. But an interesting question arises here. Ok A body goes faster but how fast does it go faster?! This is not a silly question. A hard kick given to a football will make the ball go faster fast (at once)

But if I continue to give a force slowly to the same ball for a long time ultimately it will achieve a high speed but it will have taken a longer time.

A greater force makes a body go faster, faster!

A lesser force makes a body go faster but slower when compared to a greater force!

Hence acceleration is not just change of speed but the rate at which the speed is changing i.e. how fast it is changing.

More force, faster (more quickly) the speed will change.

Lesser force, slower (more slowly) the speed will change.

You understand?

Acceleration is a key concept in Physics. So one part of Newton’s 2nd Law simply says that greater the force, greater the acceleration, lesser the force, lesser the acceleration, no force - no acceleration i.e. rest or uniform motion (Newton’s first law)!

So, in a way, Newton’s 1st law is contained in Newton’s 2nd law in an obvious way.

Now, let us come to the 2nd part of Newton’s 2nd Law.

Greater the mass, lesser the acceleration for the same force.

i.e. if certain force is applied to body A and the same force is applied to body B and if A is heavier than B.

Then which body accelerates more?

Obviously B.

In plain language, it is more difficult to change the speed of a heavier body than a lighter one. Greater force is needed to change the speed of a heavier body as compared to a lighter body.

This is plain enough.

So 2 things simultaneously determine the rate of change of speed i.e. acceleration -

1.Force, 2. Mass.

3.3. Newton’s Thrid Law

Every action (force) has an equal, opposite reaction (force) and in the same straight line.

In other words, if;

Body A gives a force to body B,

Body B gives (at once) an equal opposite force to A in the same direction.

If you push the wall, the wall pushes you back. It will be quite funny if it doesn’t! If you kick a stone, the stone will get the force but you too will get it, you’ll be hurt! Force come in pairs, there cannot be only a single force. An action force will get a reaction force at the same time. Also the reation force is opposite and exactly in a straight (opposite) line.

3.4. Fundamental forces - Gravitational, Electrical, Magnetic and Nuclear (weak and strong)

Introduction

We have said that force causes acceleration and greater the force, greater the accleration proportionately and lesser the force, lesser the acceleration. We also understand that for the same force, greater the mass, lesser the acceleration.

But a great, fundamental question still remains. Where do these forces come from? What is the origin?

Forces cannot come from nothing. There has to be something that is the cause of force, that is responsible for accelerations on masses.

Forces can come due to mass of a body (gravitational forces) - charge of a body (electrical forces) - charges in motion (magnetic forces) can originate on the nucleus of atoms (strong and weak nuclear forces)

These forces are fundamental forces. At other forces - like friction, pushes and pulls, tension on springs & strings, wind forces, muscular forces ARE AT ROOT these fundamental forces!

In the universe at large, basically, 4 forces cause all motions, cause all accelerations on masses.

These 4 forces, Gravitational, Electric, Magnetic, Nuclear forces are caused, exist due to mass, charge, charge in motion and originate in the nucleus respectively.

We will discuss electrical, magnetic and nuclear force in later sections now we will consider gravitational force.

3.5. Gravitational force or the universal law of gravitation

This force is due plainly to masses of bodies. The very mass of a body on the universe has a power - a power to attract another mass! Every mass attracts every other mass in the universe!

Greater the masses greater the force between them (and it is attractive, the masses come closer). Greater the distance, lesser the force between them. In fact it is inversely proportional to the square of the distance. If the distance is doubled, the force reduces 4 times. If the distance is tripled, the force reduces 3x3=9 times and so on..!

Gravitational force is a very weak force (compared to electrical or magnetic force). Masses attract, surely, but one of the mass atleast should be big enough for a visible acceleration.

Two chairs near each other do not attract but if one chair becomes Earth, the other chair ‘falls’ on it.

On the moon too an object falls but since the mass of the moon is 1/6 th of the Earth, the force and hence the acceleration due to gravity is 1/6 th of Earth. It falls slowly like in slow motion!

In space, if a body is left alone, nothing happens as it is very far away from any big object and negligible acceleration is observed because of negligible force. So a body in space left alone remains there! Nothing happens. It remains at rest because no force is acting on it! It simply hangs there!

Gravitational law discovered by Newton is a tremendously simple and a powerful fundamental fact in the universe. Masses attract!

The whole of space science is explained, almost all phenomena on a macroscopic level can be understood exactly, completely, beautifully by this law and it’s simple and yet the implications and application of this law staggers the mind, makes one’s breath stop!

It blows the mind to use a common phrase!