Physics for Backbenchers
Sharad B Nalawade
Physics for Backbenchers
Copyright © 2014 by Sharad B Nalawade
All rights reserved. This book or any portion th ereof
may not be reprodu ced or used in any manner whatsoever
without the express written p ermission of the publisher
except for the use of brief quotations in a book review.
First Edition, 2014
ISBN 978-93 -5174-808 -3
Book Series: Backben ch ers Series of Books ©
F002, Adarsh Residen cy, 47
th
Cross, 8
th
Blo ck, Jayan agar
Bangalore – 560070
sharadbn@gmail.com
Printers:
Lotus Printers Private Ltd., Bangalore
Table of Contents
Chapter 1 Archimedes versus Newton.............................................
Eureka! Eureka!.................................................................................
The Moon is falling ...........................................................................
Chapter 2 Pendulum Mania .............................................................
Chapter 3 Merry-go-round ...............................................................
The Turning Force ............................................................................
Chapter 4: Waves .............................................................................
Chapter 5: Let’s Calculus ..................................................................
Over to Integration ...........................................................................
Differential Equations .......................................................................
Chapter 6: The cousins .....................................................................
Chapter 7: A Hot Topic....................................................................
Chapter 8: SOH CAH TOA .............................................................
Chapter 9: The Invisible Arrows.......................................................
Friction .............................................................................................
Gravity revisited ................................................................................
Chapter 10: Relatively Speaking .........................................................
Space- time .........................................................................................
Chapter 11: Einstein’s Gravity ............................................................
Appendix: Exercises to boost your confidence .................................
List of problems: ...............................................................................
Index .....................................................................................................
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About the Author
Sharad B Nalawade writes out of his passion for science. He enjoys
communicating the most intricate concepts of science to general readers. His
first book, The Speed of Time, on the mysteries of our universe was a bestseller.
Sharad spent over 25 years in the IT industry before taking up to writing. His
love for Physics and Maths combined with his passion for making these
subjects popular among the students and the cur ious minded readers
continues to motivate him.
Sharad lives with his wife Shashi, son Akarsh and daughter Yashaswi in
Bangalore.
Acknowledgements
A book of this nature is impossible without the support from family and
friends. Young talents, Smriti and Yashaswi have helped a great deal in
shaping this book. Smriti designed the cover page truly reflecting the core
theme of the book. Yashaswi, being an undergrad in Physics herself,
relentlessly worked on the concepts providing innumerable tips. I am
thankful to both.
Vinay and Ashwini contributed immensely by reading every line of the
manuscript and correcting all my silly mistakes. They hardly slept during the
week, working nonstop to meet the deadline. I am indebted to them.
My wife, Shashi has been a great support all along the project without which
this book would never see the light of the day.
Finally, special thanks to the countless number of readers who have emailed
me with their appreciation, admiration, best wishes, suggestions, questions
and feedback. They are my greatest inspiration and reason to keep writing.
5
Preface
Being a backb en ch er in a class room is so much fun. The other day, I went to
lecture at one of the institutes and as soon as I entered the classroom, I found most
of the front rows empty with the b ack ben ch es completely full! It was a fairly big
classroom and if you are sitting at the back ben ch, your intentions are quite clear.
So I decided to have a bit fun. I went to the opposite end of the room and started
teaching! All the backb en chers were now closest to me with their backs facing me!
I was a backb en cher myself and I know all their tricks. Anyway, the experien ce was
quite enjo yable.
By backb en cher, I don’t mean th e mis chievous, noisy, bored, distracting, least
engaged and pretentious types. These qualities come and go in most of us. A
backben cher means someone who has lost interest in the subject and so the
opportunity to learn it the first time in the class room has been wasted, but later
realized that they wanted just one more chan ce. The reason to be a backben ch er
may be many. Some feel the class room to be an intimidating place for learning or
the reason could be that the teach er is either boring or bossy or ther e simply is no
motivation to learn. Whatever may be the reason, many of us miss the chan ce to
learn in a serious class room setup and later struggle to catch up with others. By the
way, Backben ch ers are smart folks!
So, the phrase backben cher here is not us ed negatively, but more circumstantially
and metaphorically. In any case, th e whole idea of this book is to present physics to
students and generally cu rious minded readers. If you are not a student anymore
but always felt that you missed the whole point about physics th en this book is
equally good for you. Students studying from high s chool all the way to post
graduation can use this book as a supplement to enhance their understanding of
physics con cepts. This book can be a good companion to guide you thro ughout
your learning and can be read along with your other text books. Although the book
starts with very simple con cepts, it takes you far into advanced con cepts and
prepares you well for solving advan ced problems!
Physics is a non-intuitive subject. Man y times, it does not appear to be a common
experien ce. You need a fair amount of visualization and thinking to understand a
con cept. Of course, not all con cepts are hard. The other aspect of physics is that it
is intimately related to mathematics. If you li ke ph ysics but not math, then there is
very little you can do about it. Math itself is a highly non -intuitive subject and when
combined with physics, the challenge may look quite daunting. Many students feel
the heat not because th ey don’t like physics but because there is so mu ch of math
in it! You will learn in this book that math is as beautiful as physics! Many times
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while reading this book, you will wonder if you are learning more math than
physics! But this much math is required to build a solid found ation in physics.
The stru cture of the book and the sequen ce of con cepts pres ented are not very
rigid and don’t compare with a standard text book! I have tried to cover most of
the basic con cepts as fully as possible. But it is n ever enough.
I have tried to make the whole experien ce o f learning physics eas y and enjoyable.
The book is narrative in its style and uses examples from our daily life as much as
possible. Remember, the whole idea of the book is to drive away the fear in you
about ones inability to learn hard con cepts in physics. Obviously, this book is not a
replacement for a good interactive teach er and if you are lu cky to have on e, keep
this book as a supplement. As a student of physics a long time ago, I was always in
search of an un conventional and ridiculously simple book and I did find one in
Physics can be fun by Y. Perelman. I must also admit that this book is highly
influen ced by the immortal Feynman Series of Lectures on Physics by one of most
brilliant physicists of all time: Richard Feynman.
Physics for Backbenchers focuses on the fundamental con cepts in physics su ch as
Newton’s laws of motion, Friction, Rotational motion, Wave nature, Thermodynamics, Einstein’s
Theory of Relativity – both Special and General Theory of Relativity , s ome math concepts like
Calculus, Vectors, Trigonometry, e and Logarithm,. What about other topics like Field
Theory, Optics, Light, Electricity & Magnetism, Atoms, Quantum Mechanics, etc .? Well,
most of these topics are extensions of the basic con cepts b eing co vered in this
book.
I have carefully s elected 20 problems and put them in the appendix that will make
you feel confid ent! On ce you are done with these problems, you are ready to take
on more ch allenges in physics! After reading this book, if readers and students get
motivated to pursue physics as their cours e at higher levels, then my purpose is
met. Fundamental and pure scien ce subjects like Physics, Biology, Chemistry and
Math will remain with us forever. I f we don’t reveal th eir inner beauty in the early
stage of student life, it would be a great loss to the society and the world as a
whole.
Alright, without much ado, let’s get cracking!
Sharad Nalawade 19
th
May, 2014
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Chapter 1 Archimedes versus Newton
Albert Einstein reportedly once said “Education is what remains after one has
forgotten everything he learned in school”. How true! As a playful child, I bet your
first encounter with physics was not in the school. Remember, your first
ride on the swing in your backyard? What about aiming a stone at a mango
from your neighbor’s tree? And your first bicycle ride? And that great
feeling of playing on the seesaw at a nearby park? The one that I enjoyed
the most was admiring the breathtaking v iew of the night sky flooded with
twinkling stars with an occasional streak of a falling meteor. While poets
expressed this beauty in the form of a poem, the budding scientists
wondered and questioned with an agitated mind. If you cho ose to ignore
and move on, you would probably be a tax consultant!
When you translate the above mentioned fun activities into a structured one
hour physics class in a school environment, all the thrill seems to fade away.
How nice would it be to have your teacher take you all to a nearby park and
play with you all and explain in simple words the laws of physics? Or even
better, get a seesaw or a slide or a swing into the class room for a demo and
I bet the backbenchers will fight for a seat at the front!
Let’s get back to the swing and play for a while. What is the first thing you
notice? Well, it goes back and forth just like a pendulum on your grandpa’s
clock. A Pendulum starts from the rest position, goes to the left and then
comes back to the rest position only to swing to the right side before
returning to the rest position again. This is one oscillation. Thus the
pendulum keeps oscillating for a very long time. Let’s call one oscillation as
one cycle. For a pendulum in the clock this cycle takes exactly one second.
The pendulum goes on making 60 cycles in one minute and 3600 cycles in 1
hour and 86400 cycles in one day and 3,15,36,000 cycles in one year and so
on! Of course, you will need to wind your clock constantly for the
pendulum to keep going. Your backyard swing also behaves exactly in the
same manner. But the fun part is that your swing always takes same time to
complete one cycle irrespective of how wide or narrow it swings! You may
give it a harder push and let it make a wider swing but even then it takes the
same time to complete one cycle. You may like to check this out next time
you play with it. Same is the case with the pendulum. Galileo first observed
this property when sitting quietly in the cathedral of Pisa watching the
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chandelier swinging overhead. But strangely enough, he also observed one
more property of the pendulum. If you now change the length of the string
that carries the bob or the chandelier, the time to complete one cycle
changes! So, one can adjust the time period of the cycle by adjusting the
length of the string of the pendulum. In fact, manufacturers of the
pendulum clocks adjust the length of the pendulum so that it takes exactly
one second for it to make one cycle.
Let’s now move to a seesaw in the park. When you play on a seesaw, you
will observe that even if your friend is heavier than you, you can still make
him go up easily by yourself sitting close to the edge of the seesaw at the
opposite end. A seesaw has a wooden plank whose center is fixed to a
fulcrum or a pivot. The length of the plank on either side of the pivot is
same. That means, when no one is playing on it, it should balance perfectly
and stay horizontal. When you and your friend sit on the opposite sides of
the seesaw, depending upo n where you sit, you will create a turning or a
lifting force. The farther away you sit fro m the center or pivot, the greater is
the lifting force you will produce! This is exactly what the great Archimedes
said. He went on to further challenge b y saying famously, “Give me a long rod,
a fulcrum and some place to stand, I will move the earth!”. Simple ideas of the great
minds!
Thus, one could probably learn and wonder at the fun rides in the park and
try to figure out if they can be explained using simple laws of physics. You
can explain for example, why the bicycle you are riding on, does not fall
sideways when it is moving or for that matter, why the moon does not fall
on the earth. Further why a stone thrown up with a sufficiently large force
never returns to the earth, or why large ships float on the water, why you
see colors in a rainbow, why electrons don’t fall into the atomic nucleus,
why our universe is expanding, why one does not age when moving at the
speed of light, why a single electron exists in two places at the same time
and so on.
Archimedes did dazzle the world by his challenge. He knew he was right
and he had faith in the laws of physics that they would never ditch him.
Even though he was one of the greatest thinkers of all time, he had no idea
about the concept of gravitation. It took over 1900 years and a genius like
Isaac Newton to discover gravity that somehow never occurred to his
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predecessors. All Newton said was that there is always a force of attraction
between two masses. For example, sun and earth are pulling at each other
with some force. As of today no one (not even Einstein) knows how exactly
this force of gravitation attracts objects at such vast distances. Newton gave
a formula to express this force of attraction between any two massive
bodies. Assume that, you and your friend are the only two people in the
entire universe and that you are floating in the vast continuum o f empty
space with no earth or sun or stars, nothing but absolute void. The only
force that now exists in the entire universe is the force between you and
your friend. Since you are the only two massive bodies next to each other,
the force of attraction between you two will keep you both close to each
other. How much is that force of attraction? Well, Newton said, the
attractive force between you two depends on how far away you are from
each other and also how large are your masses. If you are too close, this
force of attraction is greater. Likewise, if you are away from each other, this
force reduces. Okay, but by what factor? Well, assume you and your friend
are one meter apart and the force of attraction between you two is X. Now
when you move farther apart by say one meter more (you are now two
meters apart) the force will reduce by 75%. That is, it is only 25% of the
original force X. If you move farther apart, say by three meters, the force
will now reduce to only 10% of the original force X. So, you will observe
that, as you move away from each other, the force reduces rapidly. So, don’t
go too far from each other or else you will never ever be able to shake
hands again! What happens to the force of attraction if we increase the
masses of the bodies but keep the distance between them the same? Say,
you weigh 75 kilograms and yo ur friend is 100 kilograms. The force of
attraction is now proportional to 75 multiplied by 100 which is 7500 units
of force! Well almost. We will see why almost later. But I am sure this will
give us some idea about how the force of attraction depends on the masses
of the bodies and the d istance between them . Thus, more massive the
bodies the greater is the force of attraction between them and farther away
they are from each other, the lesser is the force of attraction.
I said earlier that the force of attraction (that we call force of gravitation)
between any two or more bodies depends upon their masses and the
distance between them. More precisely, the force of attraction between any
two masses depends directly upon the product of the two masses (i.e. x
) and inversely proportional to the square of the distance between the them.
10
(1/ , if ‘d’ is the distance between the masses). This is universally true.
This is Newton’s law of gravitation. You take any two masses anywhere in the
universe and measure the force between them, the above law always holds
well. One thing you must observe is that the gravitational force is always
attractive and never repulsive. Unlike in the case of magnets where you
have both the types viz. ‘Like poles’ repel each other and ‘Unlike poles’
attract each other, t here is no polarity for gravity. If you keep the two
masses and keep the distance between them fixed and take the setup to any
corner of the universe, the magnitude of the force between them is always
the same. This means that there is something unique about this force of
gravitation.
We used the terms like mass, force, etc. quite casually in the previous
paragraph but in physics they have a well defined meaning and we will study
them as we go along. One thing that a lways troubles the curious minded is
the exact difference between mass and weight. When you measure your
weight on a weighing machine at home you may say for example, it is 70
kilograms. If you now take the weighing machine to the moon and weigh
yourself again, it will now show your weight as only 11.5 kilograms! So the
best way to lose weight is to visit moon! But if you weigh yourself on the
surface of Jupiter, you will weigh as much as 165 kilograms! Is there a
difference between mass and weight? Yes there is, and the difference is that,
while the mass always remains the same the weight depends upon how
much gravitational force is acting on you. Hence, mass is something that is
much more fundamental. It is something intrinsic in you. This applies to
every matter be it a human being or a piece of stone. If you lose mass, it
gets converted into energy which is what Einstein’s famous equation
e=m says. Even while you were on the moon, your mass remained the
same but your weight got reduced. Since the gravitational force on earth is
more than that on moon, you will weigh more on the earth than on the
moon. Likewise, you will weigh more on the Jupiter than on the earth
because, Jupiter is much more massive than earth and hence it will have a
greater gravitational force. To recap, mass is the amount of matter and its
weight is decided by the gravitational force acting on it. Mass is always the
same wherever you go in the universe but not the weight. Lifting heavy
objects on the earth is more difficult than on the moon. Whenever you lift
something you have to overcome the force of gravity. Try lifting a heavy
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table and the gravitation force will pull it down with stronger force . You
can lift the same table easily on the surface of the moon.
Now the next question is: Can I measure my mass if I know my weight? Yes, you
can. Mass and its corresponding weight are related through the force of
gravity! More is th e gravitational force on a body for a given mass, more is
its weight. Before we proceed, let’s first tr y and understand what is force. In
physics, force is considered as something that brings about a change in the
body on which it is acting. What change? Well, force makes body move or
accelerate. If you want to move a body faster, you will need to apply more
force. Likewise, if the body is heavier, you will need more force to move it.
So, somehow force is related to two things: how fast you want to move a
body and how heavy is its mass. It does not matter if the force is ordinary
force of pushing or if it is a force of gravitation. If something is falling
under the influence of the gravity, it accelerates while falling down. What is
acceleration? We will learn about it shortly, but for now, let’s consider
acceleration as speed that keeps changing. If a body is moving at some
constant speed and we apply some force to it, then it will start moving
faster. Thus force induces acceleration in a body. In the case of a falling
body, the gravitational force is inducing acceleration and it is called ‘g’ in
physics.
Can you consider weight as force? Yes! If you are standing on earth, you are
being pulled down by the force of gravity all the time. This is because you
have mass. Although, we are standing still on the ground, we are under the
influence of the force of grav ity and thus undergoing acceleration ‘g’! But
we had said earlier that acceleration is something that makes an object
move faster! Then how come we are standing still? Well, this is because our
movement downward is blocked by the earth itself! We are stuck to earth so
to say. By chance, if th e land below us slides, we fall down with acceleration
g! So what we call weight is nothing but a force that is acting on us all the
time due to gravity. When I say, my weight is more than my friend’s weight,
what I mean by that is, the force acting on me is higher tha n the force
acting on my friend. This is because I am more massive than my friend. Let’s
not worry at this stage, if it does not make sense but we will soon get there.
As said, force say F is related to the mass of a body and the acceleration it
induces in it. In math, it is expressed as F=ma or F=mg depending upon
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whether it is an ordinary force or gravitational force. What’s F=mg really?
How did we get this? A force in physics is something that causes som e
change in the body or an object. We use force to push a chair, or lift a bag,
kick a ball and so on. We all know that, the amount of force that we will
need to do some activity depends upon what change we want to bring
about. For example, if we want to kick a ball, we decide to apply some force
on the ball so it starts rolling on the floor. Compare this to the force that
one needs to lift a bucket of water. In either case, the force we used has
resulted in some change. In the case of a ball, it actually m oved from one
point to another with some speed. In the case of a bucket, the force helped
us in lifting it against the force of the gravity. When we kick a ball or lift the
bucket of water, we need different amounts of force. The amount of force
that we need really depends upon the mass of the object. So in plain
English, we could say, force is something that brings about a change in the
state of an object on which force is being applied. Smaller the mass, lesser
the force we need to move it from one point to another. We can also
express this in math as force = mass x acceleration. All it means is that, the
amount of force we apply on an object depends upon its mass and how fast
we want it to move. If you are given two footballs of equal mass and you
kick them with different amounts of force, you will see that the two balls
roll with different acceleration s. Any object that moves from its resting
position has some acceleration. Acceleration is nothing but the rate of change
of velocity and velocity is the rate of change of position. Wow! Too many
terms in such a short time! Don’t worry, we will shortly study both velocity
and acceleration in more detail, but it will help us to know at this stage that,
different amounts of force induce different amounts of acceleration in an
object. If the force is greater, so is the acceleration. In short force is
something that makes an object go faster and it changes its position from
one place to another.
It is now clear that, both acceleration and mass tell us how much f orce one
needs to apply to an object in order to bring about a change in its state. We
can say, force α mass and acceleration. Here, the symbol α is a proportionality
symbol. In the language of mathematics, we say force α mass x acceleration.
If you take an object of mass 1 kg and apply force on it so it starts
accelerating with say 10 kilometers per second every second , we say, the
force we applied is equal to 1 kg x 10 km/sec/sec. So, in reality, force, F is
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nothing but the product of mass and acceleration. Hence, F = ma is our
natural conclusion!
Now let’s come to our earlier question of why force (F=ma) and weight are
one and the same. As said earlier, every matter or object in the universe has
some intrinsic mass. This mass is always under the influence of some force.
For example, our own body mass is being dragged down continuously by
the force of gravity. Even though we are being pulled by the gravity, we try
to overcome it by exerting some force of our own. If w e don’t do this, we
fall flat on the floor and remain stuck to the earth! To live and move
around, we will need to counterbalance this force of gravity all the time.
And the force that our bodies use to counterbalance the effects of gravity is
nothing but our weight! So, there is no harm in saying that weight is one
type of force. Hence, we have F=ma=mg. Why ‘g’ and not ‘a’? Well, ‘g’ is
the acceleration due to gravity so we replace ‘a’ with ‘g’. We use ‘g’ in the
case of gravity and ‘a’ in normal cases such as driving a car, etc. Even
though you are standing still on the floor your body is under a constant
force of gravity by the earth. If you were at some height instead and if you
choose to fall, you will fall w ith an acceleration ‘g’ due to gravity. When you
apply some force to move an object of mass ‘m’ with some acceleration say,
‘a’ then we say, the force = ma. But when the same object is falling down
due to the force of gravity acting on it, the object gets an acceleration ‘g’, so
we say, the force of gravity = mg. Thus F=mg is just a special case of more
general F=ma. In physics, we take ‘mg’ as weight (W), so we have
F=W=mg. If you know the value of F or W and ‘g’, you can find the value
of mass ‘m’.
As an example, if your weight is 75 kg and if you want to know your mass,
all you have to do is divide this weight by ‘g’ as F=W=mg. Obviously, your
mass is lesser than your weight by an amount proportional to ‘g’. The ‘g’ on
earth is more than the ‘g’ on the moon and hence, your weight reduces on
the moo n even though your mass is the same.
Gifting mankind with the law of gravitation was one of greatest
contributions of Newton. But he also discovered equally great laws called
the laws of motion. There are three laws of motion and they seem to be a bit
non-intuitive in nature. Let’s see in simple terms what these laws of motion
14
are and later we can try and develop a better understanding of these laws
with some examples.
First of all, it is a common experience that nothing m oves unless pushed. A
football o n the ground remains stationary for ever unless someone moves it
by kicking or a strong wind pushes it. This is intuitive. But what is not
intuitive is that, if a football is moving, it does not stop unless someone or
something stops it. But in reality, when you kick a ball, it rolls for a while
and then stops on its own. Well, this is because there is friction on the
ground that stops the moving ball. If there was no fr iction or air, the
football would move forever! This is non-intuitive. Many of the celestial
bodies like earth, moon, sun, stars, etc. keep moving in the space due to this
property. The inabi lity to stop from moving or moving without being
pushed is termed as inertia. So, Newton’s first law of motion states: “An object is
at rest or it moves at a constant velocity unless acted upon by an external force”. What
this means is that, if something is at rest, it will continue to remain at rest
until such time that someone moves it by applying some external force.
Likewise, if some object is already moving at a constant velocity, it continues to
move at that constant velocity unless someone applies some external force
to slow it down or stop it or speed it up. Well, first of all, there is a technical
term here called constant velocity. But first, what’s velocity? In physics, velocity
is the speed at which something moves. Velocity is the change in position
of an object w.r.t. time. If you cover 50 km on your bike in one hour, we
say your speed or velocity is 50 km/hour. Obviously, you cannot maintain
the same speed of 50 km/hour throughout the journey of one hour (we will
discuss the reason shortly). But for the sake of argument, let’s assume that
this is the case. We can then say that the biker is now moving with a
constant velocity. If the road on which you are biking is smooth and
straight without any curves, one can imagine the velocity to be constant.
But we all know this is not possible. What are the conditions for your bike
to be moving at a constant velocity? We just saw two conditions: the road
should be smooth without any friction and it must be straight without any
curves. Any other conditions? What if we accelerate the bike while going
with a constant velocity? Obviously, we will not be able to maintain
constant speed anymore. Another condition is that we should not apply the
brakes. If we apply the brakes while moving with a constant velocity, our
bike will slow down thus once again coming out of the constant velocity.
So, if you accelerate or brake, you can break out of constant velocity. But to
15
do these two (accelerating and braking), you will need some force. Any
external force we apply to a bike moving at a constant velocity will result in
a non constant velocity. In the absence of any external force or friction, a
body keeps moving at a constant velocity without accelerating or slowing
down. To accelerate something or to slow it down, one needs to apply a
force. Newton’s first law of motion is really all about the concept of a body
at rest or a body moving at constant velocity. Supposing, something is
moving at a constant velocity, do we need any external force to maintain it
at that constant velocity forever? No! As long as there is no friction that
stops the moving body and there are no curves, we just don’t have to worry
about any additional or external force at all. This is certainly very difficult to
visualize this in the world we live in. Our world is full of frictional forces,
curvy roads and pot holes! All that the first law of motion is saying is that, if
a car is moving along a straight road at a constant velocity (same distance
travelled in the same amount of time during the entire journey) and there is
zero friction then the car will never stop. But the moment you decide to
accelerate the car or to stop it, you will need to apply some external force.
If we kick the ball, it starts rolling on the ground with some acceleration.
The acceleration with which the ball moves depends on how hard one kicks
the ball, which is to say, how much force we apply. The more force we apply,
the faster the ball rolls. This is obvious. But o ne must also notice that, while
the acceleration depends upon the force one applies, what about the mass
of the body? Instead of a leather ball, if we are pushing a steel ball that has
much greater mass then the amount of force that we need to apply to make
it accelerate as much as an ordinary football, it would be many times more!
This means, the force we will need to apply to a body depends on its mass
as well as the acceleration we want in it. As we saw, if F is the force we are
applying on a body of mass ‘m’ and ‘a’ is th e acceleration it induces, then we
can safely say, F = ma. Whenever we have a product (multiplication) in the
formula, it always means it depends on all the quantities in the product. So,
force that we apply on a body depends on its mass and acceleration. What
about the relationship between acceleration and mass if we assume force to
be constant? This would mean that the acceleration is inversely proportional
to the mass. That is, higher the mass, lesser is the acceleration for the same
force applied. Also, lower is the mass, higher is the acceleration induced for
the sam e force. This is Newton’s second law of motion. Let’ s say it in technical
terms: “The acceleration of a body is directly proportional to, and in the same direction
16
as, the net force acting on the body and inversely proportional to its mass”. As you
noted, there is direction as well. The object always moves in the direction of
the force being applied on it. We are assuming that there is no friction. The
friction changes the direction of motion! This may not look obvious but it
is true.
And then there is Newton’s third law of motion. “ When one body exerts a force
on a second body, the second body also exerts equal and opposite force on the first body ”.
This again is a bit non-intuitive. Push the wall hard and you will feel some
pain in your shoulders. Why? Well, when you push the wall, the wall also
pushes you back! Another example is, when you fire a bullet from a gun,
you will be pushed back due to the recoil of the gun. The rocket that is fired
into space is yet another example of 3
rd
law of motion in act ion. Simplest
example is when you drop a tennis ball on the floor, it bounces back at you.
So, to sum up we have the three laws of motion and one law of gravitation.
This is the work of a genius like Newton. No wonder we all adore him so
much. With this, we are now more comfortable than ever to move on with
more interesting aspects of nature. The study of motion is a part of what is
called a Classical or Newtonian Mechanics as against Relativistic Mechanics
introduced by Einstein that we will study later in the book.
Want to reader further? Please visit amazon.in or flipkart.com to
purchase the book.
Sharad B Nalawade
Physics for Backbenchers
Copyright © 2014 by Sharad B Nalawade
All rights reserved. This book or any portion th ereof
may not be reprodu ced or used in any manner whatsoever
without the express written p ermission of the publisher
except for the use of brief quotations in a book review.
First Edition, 2014
ISBN 978-93 -5174-808 -3
Book Series: Backben ch ers Series of Books ©
F002, Adarsh Residen cy, 47
th
Cross, 8
th
Blo ck, Jayan agar
Bangalore – 560070
sharadbn@gmail.com
Printers:
Lotus Printers Private Ltd., Bangalore
Table of Contents
Chapter 1 Archimedes versus Newton.............................................
Eureka! Eureka!.................................................................................
The Moon is falling ...........................................................................
Chapter 2 Pendulum Mania .............................................................
Chapter 3 Merry-go-round ...............................................................
The Turning Force ............................................................................
Chapter 4: Waves .............................................................................
Chapter 5: Let’s Calculus ..................................................................
Over to Integration ...........................................................................
Differential Equations .......................................................................
Chapter 6: The cousins .....................................................................
Chapter 7: A Hot Topic....................................................................
Chapter 8: SOH CAH TOA .............................................................
Chapter 9: The Invisible Arrows.......................................................
Friction .............................................................................................
Gravity revisited ................................................................................
Chapter 10: Relatively Speaking .........................................................
Space- time .........................................................................................
Chapter 11: Einstein’s Gravity ............................................................
Appendix: Exercises to boost your confidence .................................
List of problems: ...............................................................................
Index .....................................................................................................
4
About the Author
Sharad B Nalawade writes out of his passion for science. He enjoys
communicating the most intricate concepts of science to general readers. His
first book, The Speed of Time, on the mysteries of our universe was a bestseller.
Sharad spent over 25 years in the IT industry before taking up to writing. His
love for Physics and Maths combined with his passion for making these
subjects popular among the students and the cur ious minded readers
continues to motivate him.
Sharad lives with his wife Shashi, son Akarsh and daughter Yashaswi in
Bangalore.
Acknowledgements
A book of this nature is impossible without the support from family and
friends. Young talents, Smriti and Yashaswi have helped a great deal in
shaping this book. Smriti designed the cover page truly reflecting the core
theme of the book. Yashaswi, being an undergrad in Physics herself,
relentlessly worked on the concepts providing innumerable tips. I am
thankful to both.
Vinay and Ashwini contributed immensely by reading every line of the
manuscript and correcting all my silly mistakes. They hardly slept during the
week, working nonstop to meet the deadline. I am indebted to them.
My wife, Shashi has been a great support all along the project without which
this book would never see the light of the day.
Finally, special thanks to the countless number of readers who have emailed
me with their appreciation, admiration, best wishes, suggestions, questions
and feedback. They are my greatest inspiration and reason to keep writing.
5
Preface
Being a backb en ch er in a class room is so much fun. The other day, I went to
lecture at one of the institutes and as soon as I entered the classroom, I found most
of the front rows empty with the b ack ben ch es completely full! It was a fairly big
classroom and if you are sitting at the back ben ch, your intentions are quite clear.
So I decided to have a bit fun. I went to the opposite end of the room and started
teaching! All the backb en chers were now closest to me with their backs facing me!
I was a backb en cher myself and I know all their tricks. Anyway, the experien ce was
quite enjo yable.
By backb en cher, I don’t mean th e mis chievous, noisy, bored, distracting, least
engaged and pretentious types. These qualities come and go in most of us. A
backben cher means someone who has lost interest in the subject and so the
opportunity to learn it the first time in the class room has been wasted, but later
realized that they wanted just one more chan ce. The reason to be a backben ch er
may be many. Some feel the class room to be an intimidating place for learning or
the reason could be that the teach er is either boring or bossy or ther e simply is no
motivation to learn. Whatever may be the reason, many of us miss the chan ce to
learn in a serious class room setup and later struggle to catch up with others. By the
way, Backben ch ers are smart folks!
So, the phrase backben cher here is not us ed negatively, but more circumstantially
and metaphorically. In any case, th e whole idea of this book is to present physics to
students and generally cu rious minded readers. If you are not a student anymore
but always felt that you missed the whole point about physics th en this book is
equally good for you. Students studying from high s chool all the way to post
graduation can use this book as a supplement to enhance their understanding of
physics con cepts. This book can be a good companion to guide you thro ughout
your learning and can be read along with your other text books. Although the book
starts with very simple con cepts, it takes you far into advanced con cepts and
prepares you well for solving advan ced problems!
Physics is a non-intuitive subject. Man y times, it does not appear to be a common
experien ce. You need a fair amount of visualization and thinking to understand a
con cept. Of course, not all con cepts are hard. The other aspect of physics is that it
is intimately related to mathematics. If you li ke ph ysics but not math, then there is
very little you can do about it. Math itself is a highly non -intuitive subject and when
combined with physics, the challenge may look quite daunting. Many students feel
the heat not because th ey don’t like physics but because there is so mu ch of math
in it! You will learn in this book that math is as beautiful as physics! Many times
6
while reading this book, you will wonder if you are learning more math than
physics! But this much math is required to build a solid found ation in physics.
The stru cture of the book and the sequen ce of con cepts pres ented are not very
rigid and don’t compare with a standard text book! I have tried to cover most of
the basic con cepts as fully as possible. But it is n ever enough.
I have tried to make the whole experien ce o f learning physics eas y and enjoyable.
The book is narrative in its style and uses examples from our daily life as much as
possible. Remember, the whole idea of the book is to drive away the fear in you
about ones inability to learn hard con cepts in physics. Obviously, this book is not a
replacement for a good interactive teach er and if you are lu cky to have on e, keep
this book as a supplement. As a student of physics a long time ago, I was always in
search of an un conventional and ridiculously simple book and I did find one in
Physics can be fun by Y. Perelman. I must also admit that this book is highly
influen ced by the immortal Feynman Series of Lectures on Physics by one of most
brilliant physicists of all time: Richard Feynman.
Physics for Backbenchers focuses on the fundamental con cepts in physics su ch as
Newton’s laws of motion, Friction, Rotational motion, Wave nature, Thermodynamics, Einstein’s
Theory of Relativity – both Special and General Theory of Relativity , s ome math concepts like
Calculus, Vectors, Trigonometry, e and Logarithm,. What about other topics like Field
Theory, Optics, Light, Electricity & Magnetism, Atoms, Quantum Mechanics, etc .? Well,
most of these topics are extensions of the basic con cepts b eing co vered in this
book.
I have carefully s elected 20 problems and put them in the appendix that will make
you feel confid ent! On ce you are done with these problems, you are ready to take
on more ch allenges in physics! After reading this book, if readers and students get
motivated to pursue physics as their cours e at higher levels, then my purpose is
met. Fundamental and pure scien ce subjects like Physics, Biology, Chemistry and
Math will remain with us forever. I f we don’t reveal th eir inner beauty in the early
stage of student life, it would be a great loss to the society and the world as a
whole.
Alright, without much ado, let’s get cracking!
Sharad Nalawade 19
th
May, 2014
7
Chapter 1 Archimedes versus Newton
Albert Einstein reportedly once said “Education is what remains after one has
forgotten everything he learned in school”. How true! As a playful child, I bet your
first encounter with physics was not in the school. Remember, your first
ride on the swing in your backyard? What about aiming a stone at a mango
from your neighbor’s tree? And your first bicycle ride? And that great
feeling of playing on the seesaw at a nearby park? The one that I enjoyed
the most was admiring the breathtaking v iew of the night sky flooded with
twinkling stars with an occasional streak of a falling meteor. While poets
expressed this beauty in the form of a poem, the budding scientists
wondered and questioned with an agitated mind. If you cho ose to ignore
and move on, you would probably be a tax consultant!
When you translate the above mentioned fun activities into a structured one
hour physics class in a school environment, all the thrill seems to fade away.
How nice would it be to have your teacher take you all to a nearby park and
play with you all and explain in simple words the laws of physics? Or even
better, get a seesaw or a slide or a swing into the class room for a demo and
I bet the backbenchers will fight for a seat at the front!
Let’s get back to the swing and play for a while. What is the first thing you
notice? Well, it goes back and forth just like a pendulum on your grandpa’s
clock. A Pendulum starts from the rest position, goes to the left and then
comes back to the rest position only to swing to the right side before
returning to the rest position again. This is one oscillation. Thus the
pendulum keeps oscillating for a very long time. Let’s call one oscillation as
one cycle. For a pendulum in the clock this cycle takes exactly one second.
The pendulum goes on making 60 cycles in one minute and 3600 cycles in 1
hour and 86400 cycles in one day and 3,15,36,000 cycles in one year and so
on! Of course, you will need to wind your clock constantly for the
pendulum to keep going. Your backyard swing also behaves exactly in the
same manner. But the fun part is that your swing always takes same time to
complete one cycle irrespective of how wide or narrow it swings! You may
give it a harder push and let it make a wider swing but even then it takes the
same time to complete one cycle. You may like to check this out next time
you play with it. Same is the case with the pendulum. Galileo first observed
this property when sitting quietly in the cathedral of Pisa watching the
8
chandelier swinging overhead. But strangely enough, he also observed one
more property of the pendulum. If you now change the length of the string
that carries the bob or the chandelier, the time to complete one cycle
changes! So, one can adjust the time period of the cycle by adjusting the
length of the string of the pendulum. In fact, manufacturers of the
pendulum clocks adjust the length of the pendulum so that it takes exactly
one second for it to make one cycle.
Let’s now move to a seesaw in the park. When you play on a seesaw, you
will observe that even if your friend is heavier than you, you can still make
him go up easily by yourself sitting close to the edge of the seesaw at the
opposite end. A seesaw has a wooden plank whose center is fixed to a
fulcrum or a pivot. The length of the plank on either side of the pivot is
same. That means, when no one is playing on it, it should balance perfectly
and stay horizontal. When you and your friend sit on the opposite sides of
the seesaw, depending upo n where you sit, you will create a turning or a
lifting force. The farther away you sit fro m the center or pivot, the greater is
the lifting force you will produce! This is exactly what the great Archimedes
said. He went on to further challenge b y saying famously, “Give me a long rod,
a fulcrum and some place to stand, I will move the earth!”. Simple ideas of the great
minds!
Thus, one could probably learn and wonder at the fun rides in the park and
try to figure out if they can be explained using simple laws of physics. You
can explain for example, why the bicycle you are riding on, does not fall
sideways when it is moving or for that matter, why the moon does not fall
on the earth. Further why a stone thrown up with a sufficiently large force
never returns to the earth, or why large ships float on the water, why you
see colors in a rainbow, why electrons don’t fall into the atomic nucleus,
why our universe is expanding, why one does not age when moving at the
speed of light, why a single electron exists in two places at the same time
and so on.
Archimedes did dazzle the world by his challenge. He knew he was right
and he had faith in the laws of physics that they would never ditch him.
Even though he was one of the greatest thinkers of all time, he had no idea
about the concept of gravitation. It took over 1900 years and a genius like
Isaac Newton to discover gravity that somehow never occurred to his
9
predecessors. All Newton said was that there is always a force of attraction
between two masses. For example, sun and earth are pulling at each other
with some force. As of today no one (not even Einstein) knows how exactly
this force of gravitation attracts objects at such vast distances. Newton gave
a formula to express this force of attraction between any two massive
bodies. Assume that, you and your friend are the only two people in the
entire universe and that you are floating in the vast continuum o f empty
space with no earth or sun or stars, nothing but absolute void. The only
force that now exists in the entire universe is the force between you and
your friend. Since you are the only two massive bodies next to each other,
the force of attraction between you two will keep you both close to each
other. How much is that force of attraction? Well, Newton said, the
attractive force between you two depends on how far away you are from
each other and also how large are your masses. If you are too close, this
force of attraction is greater. Likewise, if you are away from each other, this
force reduces. Okay, but by what factor? Well, assume you and your friend
are one meter apart and the force of attraction between you two is X. Now
when you move farther apart by say one meter more (you are now two
meters apart) the force will reduce by 75%. That is, it is only 25% of the
original force X. If you move farther apart, say by three meters, the force
will now reduce to only 10% of the original force X. So, you will observe
that, as you move away from each other, the force reduces rapidly. So, don’t
go too far from each other or else you will never ever be able to shake
hands again! What happens to the force of attraction if we increase the
masses of the bodies but keep the distance between them the same? Say,
you weigh 75 kilograms and yo ur friend is 100 kilograms. The force of
attraction is now proportional to 75 multiplied by 100 which is 7500 units
of force! Well almost. We will see why almost later. But I am sure this will
give us some idea about how the force of attraction depends on the masses
of the bodies and the d istance between them . Thus, more massive the
bodies the greater is the force of attraction between them and farther away
they are from each other, the lesser is the force of attraction.
I said earlier that the force of attraction (that we call force of gravitation)
between any two or more bodies depends upon their masses and the
distance between them. More precisely, the force of attraction between any
two masses depends directly upon the product of the two masses (i.e. x
) and inversely proportional to the square of the distance between the them.
10
(1/ , if ‘d’ is the distance between the masses). This is universally true.
This is Newton’s law of gravitation. You take any two masses anywhere in the
universe and measure the force between them, the above law always holds
well. One thing you must observe is that the gravitational force is always
attractive and never repulsive. Unlike in the case of magnets where you
have both the types viz. ‘Like poles’ repel each other and ‘Unlike poles’
attract each other, t here is no polarity for gravity. If you keep the two
masses and keep the distance between them fixed and take the setup to any
corner of the universe, the magnitude of the force between them is always
the same. This means that there is something unique about this force of
gravitation.
We used the terms like mass, force, etc. quite casually in the previous
paragraph but in physics they have a well defined meaning and we will study
them as we go along. One thing that a lways troubles the curious minded is
the exact difference between mass and weight. When you measure your
weight on a weighing machine at home you may say for example, it is 70
kilograms. If you now take the weighing machine to the moon and weigh
yourself again, it will now show your weight as only 11.5 kilograms! So the
best way to lose weight is to visit moon! But if you weigh yourself on the
surface of Jupiter, you will weigh as much as 165 kilograms! Is there a
difference between mass and weight? Yes there is, and the difference is that,
while the mass always remains the same the weight depends upon how
much gravitational force is acting on you. Hence, mass is something that is
much more fundamental. It is something intrinsic in you. This applies to
every matter be it a human being or a piece of stone. If you lose mass, it
gets converted into energy which is what Einstein’s famous equation
e=m says. Even while you were on the moon, your mass remained the
same but your weight got reduced. Since the gravitational force on earth is
more than that on moon, you will weigh more on the earth than on the
moon. Likewise, you will weigh more on the Jupiter than on the earth
because, Jupiter is much more massive than earth and hence it will have a
greater gravitational force. To recap, mass is the amount of matter and its
weight is decided by the gravitational force acting on it. Mass is always the
same wherever you go in the universe but not the weight. Lifting heavy
objects on the earth is more difficult than on the moon. Whenever you lift
something you have to overcome the force of gravity. Try lifting a heavy
11
table and the gravitation force will pull it down with stronger force . You
can lift the same table easily on the surface of the moon.
Now the next question is: Can I measure my mass if I know my weight? Yes, you
can. Mass and its corresponding weight are related through the force of
gravity! More is th e gravitational force on a body for a given mass, more is
its weight. Before we proceed, let’s first tr y and understand what is force. In
physics, force is considered as something that brings about a change in the
body on which it is acting. What change? Well, force makes body move or
accelerate. If you want to move a body faster, you will need to apply more
force. Likewise, if the body is heavier, you will need more force to move it.
So, somehow force is related to two things: how fast you want to move a
body and how heavy is its mass. It does not matter if the force is ordinary
force of pushing or if it is a force of gravitation. If something is falling
under the influence of the gravity, it accelerates while falling down. What is
acceleration? We will learn about it shortly, but for now, let’s consider
acceleration as speed that keeps changing. If a body is moving at some
constant speed and we apply some force to it, then it will start moving
faster. Thus force induces acceleration in a body. In the case of a falling
body, the gravitational force is inducing acceleration and it is called ‘g’ in
physics.
Can you consider weight as force? Yes! If you are standing on earth, you are
being pulled down by the force of gravity all the time. This is because you
have mass. Although, we are standing still on the ground, we are under the
influence of the force of grav ity and thus undergoing acceleration ‘g’! But
we had said earlier that acceleration is something that makes an object
move faster! Then how come we are standing still? Well, this is because our
movement downward is blocked by the earth itself! We are stuck to earth so
to say. By chance, if th e land below us slides, we fall down with acceleration
g! So what we call weight is nothing but a force that is acting on us all the
time due to gravity. When I say, my weight is more than my friend’s weight,
what I mean by that is, the force acting on me is higher tha n the force
acting on my friend. This is because I am more massive than my friend. Let’s
not worry at this stage, if it does not make sense but we will soon get there.
As said, force say F is related to the mass of a body and the acceleration it
induces in it. In math, it is expressed as F=ma or F=mg depending upon
12
whether it is an ordinary force or gravitational force. What’s F=mg really?
How did we get this? A force in physics is something that causes som e
change in the body or an object. We use force to push a chair, or lift a bag,
kick a ball and so on. We all know that, the amount of force that we will
need to do some activity depends upon what change we want to bring
about. For example, if we want to kick a ball, we decide to apply some force
on the ball so it starts rolling on the floor. Compare this to the force that
one needs to lift a bucket of water. In either case, the force we used has
resulted in some change. In the case of a ball, it actually m oved from one
point to another with some speed. In the case of a bucket, the force helped
us in lifting it against the force of the gravity. When we kick a ball or lift the
bucket of water, we need different amounts of force. The amount of force
that we need really depends upon the mass of the object. So in plain
English, we could say, force is something that brings about a change in the
state of an object on which force is being applied. Smaller the mass, lesser
the force we need to move it from one point to another. We can also
express this in math as force = mass x acceleration. All it means is that, the
amount of force we apply on an object depends upon its mass and how fast
we want it to move. If you are given two footballs of equal mass and you
kick them with different amounts of force, you will see that the two balls
roll with different acceleration s. Any object that moves from its resting
position has some acceleration. Acceleration is nothing but the rate of change
of velocity and velocity is the rate of change of position. Wow! Too many
terms in such a short time! Don’t worry, we will shortly study both velocity
and acceleration in more detail, but it will help us to know at this stage that,
different amounts of force induce different amounts of acceleration in an
object. If the force is greater, so is the acceleration. In short force is
something that makes an object go faster and it changes its position from
one place to another.
It is now clear that, both acceleration and mass tell us how much f orce one
needs to apply to an object in order to bring about a change in its state. We
can say, force α mass and acceleration. Here, the symbol α is a proportionality
symbol. In the language of mathematics, we say force α mass x acceleration.
If you take an object of mass 1 kg and apply force on it so it starts
accelerating with say 10 kilometers per second every second , we say, the
force we applied is equal to 1 kg x 10 km/sec/sec. So, in reality, force, F is
13
nothing but the product of mass and acceleration. Hence, F = ma is our
natural conclusion!
Now let’s come to our earlier question of why force (F=ma) and weight are
one and the same. As said earlier, every matter or object in the universe has
some intrinsic mass. This mass is always under the influence of some force.
For example, our own body mass is being dragged down continuously by
the force of gravity. Even though we are being pulled by the gravity, we try
to overcome it by exerting some force of our own. If w e don’t do this, we
fall flat on the floor and remain stuck to the earth! To live and move
around, we will need to counterbalance this force of gravity all the time.
And the force that our bodies use to counterbalance the effects of gravity is
nothing but our weight! So, there is no harm in saying that weight is one
type of force. Hence, we have F=ma=mg. Why ‘g’ and not ‘a’? Well, ‘g’ is
the acceleration due to gravity so we replace ‘a’ with ‘g’. We use ‘g’ in the
case of gravity and ‘a’ in normal cases such as driving a car, etc. Even
though you are standing still on the floor your body is under a constant
force of gravity by the earth. If you were at some height instead and if you
choose to fall, you will fall w ith an acceleration ‘g’ due to gravity. When you
apply some force to move an object of mass ‘m’ with some acceleration say,
‘a’ then we say, the force = ma. But when the same object is falling down
due to the force of gravity acting on it, the object gets an acceleration ‘g’, so
we say, the force of gravity = mg. Thus F=mg is just a special case of more
general F=ma. In physics, we take ‘mg’ as weight (W), so we have
F=W=mg. If you know the value of F or W and ‘g’, you can find the value
of mass ‘m’.
As an example, if your weight is 75 kg and if you want to know your mass,
all you have to do is divide this weight by ‘g’ as F=W=mg. Obviously, your
mass is lesser than your weight by an amount proportional to ‘g’. The ‘g’ on
earth is more than the ‘g’ on the moon and hence, your weight reduces on
the moo n even though your mass is the same.
Gifting mankind with the law of gravitation was one of greatest
contributions of Newton. But he also discovered equally great laws called
the laws of motion. There are three laws of motion and they seem to be a bit
non-intuitive in nature. Let’s see in simple terms what these laws of motion
14
are and later we can try and develop a better understanding of these laws
with some examples.
First of all, it is a common experience that nothing m oves unless pushed. A
football o n the ground remains stationary for ever unless someone moves it
by kicking or a strong wind pushes it. This is intuitive. But what is not
intuitive is that, if a football is moving, it does not stop unless someone or
something stops it. But in reality, when you kick a ball, it rolls for a while
and then stops on its own. Well, this is because there is friction on the
ground that stops the moving ball. If there was no fr iction or air, the
football would move forever! This is non-intuitive. Many of the celestial
bodies like earth, moon, sun, stars, etc. keep moving in the space due to this
property. The inabi lity to stop from moving or moving without being
pushed is termed as inertia. So, Newton’s first law of motion states: “An object is
at rest or it moves at a constant velocity unless acted upon by an external force”. What
this means is that, if something is at rest, it will continue to remain at rest
until such time that someone moves it by applying some external force.
Likewise, if some object is already moving at a constant velocity, it continues to
move at that constant velocity unless someone applies some external force
to slow it down or stop it or speed it up. Well, first of all, there is a technical
term here called constant velocity. But first, what’s velocity? In physics, velocity
is the speed at which something moves. Velocity is the change in position
of an object w.r.t. time. If you cover 50 km on your bike in one hour, we
say your speed or velocity is 50 km/hour. Obviously, you cannot maintain
the same speed of 50 km/hour throughout the journey of one hour (we will
discuss the reason shortly). But for the sake of argument, let’s assume that
this is the case. We can then say that the biker is now moving with a
constant velocity. If the road on which you are biking is smooth and
straight without any curves, one can imagine the velocity to be constant.
But we all know this is not possible. What are the conditions for your bike
to be moving at a constant velocity? We just saw two conditions: the road
should be smooth without any friction and it must be straight without any
curves. Any other conditions? What if we accelerate the bike while going
with a constant velocity? Obviously, we will not be able to maintain
constant speed anymore. Another condition is that we should not apply the
brakes. If we apply the brakes while moving with a constant velocity, our
bike will slow down thus once again coming out of the constant velocity.
So, if you accelerate or brake, you can break out of constant velocity. But to
15
do these two (accelerating and braking), you will need some force. Any
external force we apply to a bike moving at a constant velocity will result in
a non constant velocity. In the absence of any external force or friction, a
body keeps moving at a constant velocity without accelerating or slowing
down. To accelerate something or to slow it down, one needs to apply a
force. Newton’s first law of motion is really all about the concept of a body
at rest or a body moving at constant velocity. Supposing, something is
moving at a constant velocity, do we need any external force to maintain it
at that constant velocity forever? No! As long as there is no friction that
stops the moving body and there are no curves, we just don’t have to worry
about any additional or external force at all. This is certainly very difficult to
visualize this in the world we live in. Our world is full of frictional forces,
curvy roads and pot holes! All that the first law of motion is saying is that, if
a car is moving along a straight road at a constant velocity (same distance
travelled in the same amount of time during the entire journey) and there is
zero friction then the car will never stop. But the moment you decide to
accelerate the car or to stop it, you will need to apply some external force.
If we kick the ball, it starts rolling on the ground with some acceleration.
The acceleration with which the ball moves depends on how hard one kicks
the ball, which is to say, how much force we apply. The more force we apply,
the faster the ball rolls. This is obvious. But o ne must also notice that, while
the acceleration depends upon the force one applies, what about the mass
of the body? Instead of a leather ball, if we are pushing a steel ball that has
much greater mass then the amount of force that we need to apply to make
it accelerate as much as an ordinary football, it would be many times more!
This means, the force we will need to apply to a body depends on its mass
as well as the acceleration we want in it. As we saw, if F is the force we are
applying on a body of mass ‘m’ and ‘a’ is th e acceleration it induces, then we
can safely say, F = ma. Whenever we have a product (multiplication) in the
formula, it always means it depends on all the quantities in the product. So,
force that we apply on a body depends on its mass and acceleration. What
about the relationship between acceleration and mass if we assume force to
be constant? This would mean that the acceleration is inversely proportional
to the mass. That is, higher the mass, lesser is the acceleration for the same
force applied. Also, lower is the mass, higher is the acceleration induced for
the sam e force. This is Newton’s second law of motion. Let’ s say it in technical
terms: “The acceleration of a body is directly proportional to, and in the same direction
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as, the net force acting on the body and inversely proportional to its mass”. As you
noted, there is direction as well. The object always moves in the direction of
the force being applied on it. We are assuming that there is no friction. The
friction changes the direction of motion! This may not look obvious but it
is true.
And then there is Newton’s third law of motion. “ When one body exerts a force
on a second body, the second body also exerts equal and opposite force on the first body ”.
This again is a bit non-intuitive. Push the wall hard and you will feel some
pain in your shoulders. Why? Well, when you push the wall, the wall also
pushes you back! Another example is, when you fire a bullet from a gun,
you will be pushed back due to the recoil of the gun. The rocket that is fired
into space is yet another example of 3
rd
law of motion in act ion. Simplest
example is when you drop a tennis ball on the floor, it bounces back at you.
So, to sum up we have the three laws of motion and one law of gravitation.
This is the work of a genius like Newton. No wonder we all adore him so
much. With this, we are now more comfortable than ever to move on with
more interesting aspects of nature. The study of motion is a part of what is
called a Classical or Newtonian Mechanics as against Relativistic Mechanics
introduced by Einstein that we will study later in the book.
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