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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY
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UNIT - I
Review of the three laws of motion and vector algebra
In this course on Engineering Mechanics, we shall be learning about mechanical interaction
between bodies. That is we will learn how different bodies apply forces on one another and how
they then balance to keep each other in equilibrium. That will be done in the first part of the
course. So in the first part we will be dealing with STATICS. In the second part we then go to
the motion of particles and see how does the motion of particles get affected when a force is
applied on them. We will first deal with single particles and will then move on to describe the
motion of rigid bodies.
The basis of all solutions to mechanics problems are the Newton's laws of motion in one form or
the other. The laws are:
First law: A body does not change its state of motion unless acted upon by a force. This law is
based on observations but in addition it also defines an inertial frame . By definition an inertial
frame is that in which a body does not change its state of motion unless acted upon by a force.
For example to a very good approximation a frame fixed in a room is an inertial frame for
motion of balls/ objects in that room. On the other hand if you are sitting in a train that is
accelerating, you will see that objects outside are changing their speed without any apparent
force. Then the motion of objects outside is changing without any force. The train is a noninertial frame.
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Second law: The second law is also part definition and part observation. It gives the force in
terms of a quantity called the mass and the acceleration of a particle. It says that a force of
magnitude F applied on a particle gives it an acceleration a proportional to the force. In other
words
F = ma ,
(1)
wherem is identified as the inertial mass of the body. So if the same force - applied either by a
spring stretched or compressed to the same length - acting on two different particles produces
accelerations a1 and a2, we can say that
m1 a1 = m2 a2
or
(2)
Thus by comparing accelerations of a particle and of a standard mass (unit mass) when the same
force is applied on each one them we get the mass of that particle. Thus gives us the definition of
mass. It also gives us how to measure the force via the equation F = ma. One Newton
(abbreviated as N) of force is that providing an acceleration of 1m/s2 to a standard mass of 1 kg.
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If you want to feel how much in 1 Newton , hold your palm horizontally and put a hundred gram
weight on it; the force that you feel is about 1N.
Of course you cannot always measure the force applied by accelerating objects. For example if
you are pushing a wall, how much force you are applying cannot be measured by observing the
acceleration of the wall because the wall is not moving. However once we have adopted a
measure of force, we can always measure it by comparing the force applied in some other
situation.
In the first part of the course i.e. Statics we consider only equilibrium situations. We will
therefore not be looking at F = ma but rather at the balance of different forces applied on a
system. In the second part - Dynamics - we will be applying F = ma extensively.
Third Law: Newton's third law states that if a body A applies a force F on body B , then B also
applies an equal and opposite force on A . (Forces do not cancel such other as they are acting on
two different objects)
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Figure 1
Thus if they start from the position of rest A and B will tend to move in opposite directions. You
may ask: if A andB are experiencing equal and opposite force, why do they not cancel each
other? This is because - as stated above - the forces are acting on two different objects. We shall
be using this law a lot both in static as well as in dynamics.
After this preliminary introduction to what we will be doing in the coming lectures, we begin
with a review of vectors because the quantities like force, velocities are all vectors and we should
therefore know how to work with the vectors. I am sure you have learnt some basic
manipulations with vectors in your 12th grade so this lectures is essentially to recapitulate on
what you have learnt and also introduce you to one or two new concepts.
You have learnt in the past is that vectors are quantity which have both a magnitude and a
direction in contrast to scalar quantities that are specified by their magnitude only. Thus a
quantity like force is a vector quantity because when I tell someone that I am applying Xamount of force, by itself it is not meaningful unless I also specify in which direction I am
applying this force. Similarly when I ask you where your friend's house is you can't just tell me
that it is some 500 meters far. You will also have to tell me that it is 500 meters to the north or
300 meters to the east and four hundred meters to the north from here. Without formally
realizing it, you are telling me a about a vector quantity. Thus quantities like displacement,
velocity, acceleration, force are vectors. On the other hand the quantities distance, speed and
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energy are scalar quantities. In the following we discuss the algebra involving vector quantities.
We begin with a discussion of the equality of vectors.
Equality of Vectors: Since a vector is defined by the direction and magnitude, two vectors are
equal if they have the same magnitude and direction. Thus in figure 2 vector
and but not equal to vector
is equal to vector
although all of them have the same magnitude.
Thus we conclude that any two vectors which have the same magnitude and are parallel to each
other are equal. If they are not parallel then they cannot be equal no matter what their magnitude.
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In physical situations even two equal vectors may produce different effects depending on where
they are located. For example take the force applied on a disc. If applied on the rim it rotates
the wheel at a speed different from when it is applied to a point nearer to the center. Thus
although it is the same force, applied at different points it produces different effects. On the other
hand, imagine a thin rope wrapped on a wheel and being pulled out horizontally from the top. On
the rope no matter where the force is applied, the effect is the same. Similarly we may push the
wheel by applying the same force at thee end of a stick with same result (see figure 3).
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Thus we observe that a force applied anywhere along its line of applications produces the same
effect. This is known as transmissibility of force. On the other hand if the same force is applied
at a point away from its line of application, the effect produced is different. Thus the
transmissibility does not mean that force can be applied anywhere to produce the same effect but
only at any point on its line of application.
Adding and subtracting two vectors (Graphical Method): When we add two vectors
by graphical method to get
we draw a vector from the tail of
, we take vector
to the head of
, put the tail of
4
.Then
. That vector represents the resultant
(Figure 4). I leave it as an exercise for you to show that
that vector addition is commutative.
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on the head of
and
. In other words, show
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