Newton's laws of motion are of central importance in classical physics. A large number of principles and results may be derived from Newton's laws. The first two laws relate to the type of motion of a system that results from a given set of forces. These laws may be interpreted in a variety of ways and it is slightly uninteresting and annoying at the outset to go into the technical details of the interpretation. The precise definitions of mass, force and acceleration should be given before we relate) them. And these definitions themselves need use of Newton's laws.
First Law
of Motion
Every body continues to be in a state of rest or uniform motion in a
straight line, except in so far as it may be compelled by force to change that
state.’ Newton’s first law of motion defines inertia.
1. Inertia of Rest : the inability of
a body to change by itself its state of rest.
Ø When a branch of a fruit tree is shaken, the fruits fall down. This is
because the branch comes in motion and the fruits tend to remain at rest.
Hence, they get detached.
Ø The dirt particles in a durree fall off if it is stricken by a stick.
This is because the striking sets the durree in motion whereas the
dirt-particles tend to remain at rest and hence fall.
Ø When a train starts suddenly, the passenger sitting inside tends to fall
backwards. This is so because the lower part of the passenger’s body starts
moving with the train but the upper part tends to remain at rest.
Ø If a smooth paper having a coin on it placed on a table is suddenly
drawn, the coin remains at the same place on the table due to inertia of rest.
Ø When a horse starts suddenly, the rider tends to fall backwards due to
inertia of rest
2. Intertia of Motion : The inability of
a body to change by itself its state of uniform motion. When a horse at full
gallop stops suddenly, the rider on it falls forward because of inertia of
motion of the upper
part of the rider’s body.
Ø When an athelete takes a long jump, he runs first for a certain distance
before the jump. This is because his feet come to rest on touching the ground
and the remaining body continues to move owing to inertia of motion.
Ø When train stops suddenly, a passenger sitting inside tends to fall
forward. It happens because the lower part of the passenger’s body comes to
rest with the train but the upper part tends to continue its motion due to
inertia of motion.
Ø A person jumping out of a speeding train may fall forward due to inertia
of motion of his body. Hence, he should run a few steps on the platform in the direction
of motion of train.
3. Inertia of Direction : The inability of
a body to change by itself its direction of motion.
Ø The wheels of any moving vehicle throw out mud, if any, tangentially, due
to the inertia of direction. The mud guards over the wheels stop this mud,
protecting the clothes, etc. of the person sitting on the bike.
Ø Use of an umbrella to protect us fromrain is based on the property of
inertia of direction because the rain drops cannot change their direction of
motion.
Ø When a bus or a car rounds a curve suddenly, the person sitting inside
is thrown outwards. It happens so because the person tries to maintain his direction
of motion due to directional inertia while the vehicle turns.
Ø When a knife is sharpened by pressing it against a grinding stone, the sparks
fly off tangentially because of the inertia of direction.
Ø When a stone tied to one end of a string is whirled and the string
breaks suddenly, the stone spins off along the tangent of its circular path. It
happens so because of the pull in the string was forcing the stone to move in a
circle. As soon as the string breaks, the pull disappears. The stone becomes
free and in a bid to move along the straight line flies off tangentially.
A frame of reference in which Newton's first law is valid is called an inertial
frame of reference. A frame in which Newton's first law is not valid is
called a noninertial frame of reference. Newton's first law, thus,
reduces to a definition of inertial frame. Why do we call it a law then ? Suppose
after going through this lesson, you keep the book on your table fixed rigidly
with the earth(Figure 1)
Figure 1
The book is at rest with respect to the earth. The acceleration of the
book with respect to the earth is zero. The forces on the book are (a) the
gravitational force W exerted by the earth and (b) the cojitact force N by the
table. Is the sum of W and N zero ? A very accurate measurement will give the
answer "No". The sum of the forces is not zero although the book is at
rest. The earth is not strictly an inertial frame. However, the sum is not too
different from zero and we can say that the earth is an inertial frame of reference
to a good approximation. Thus, for routine affairs, "a = 0 if and only if
F = 0" is true in the earth frame of reference. This fact was
identified and
formulated by Newton and is known as Newton's first law. If we
restrict that all measurements will be made from the earth frame, indeed
it becomes a law. If we try to universalise this to different frames, it
becomes a definition. We shall assume that unless stated otherwise,
we are working from an inertial frame of reference.
SECOND LAW OF MOTION
The acceleration of a particle as measured from an Second Law of Motion ‘The rate of change of linear momentum of a body
is directly proportional to the external force applied on the body and this
change takes place always in the direction of the applied force’. The second
law gives us a measure of force. When a force is applied on a body, its momentum
and hence, velocity change. The change in velocity produces an acceleration in the
body. The rate of change of linear momentum with time is equal to the product of
the mass of the body and its acceleration which measures the magnitude of the
applied force i.e.
When a body is moving with a uniform velocity along a straight line, it
neither experience nor require an external force. This is because, the acceleration
is due to change in the velocity of the body and the velocity remains constant
for a body moving with a uniform velocity along a straight line. When a body
changes its velocity or direction of its motion, its velocity changes too. It results
in an acceleration which is possible only by the action of an external applied
force. Hence, an accelerated motion is always due to an external force.
Application of the change in linear momentum (impluse) and second law of
motion :
Ø Bogies of a train are provided with the buffers. These buffers avoid
severe jerks during shunting of the train. Since force = change in
momentum/time and the time of impact increases due to presence of buffers.
Hence, force during jerks decrease. It results in decrease in the chances of
damage.
Ø Crockery items are wrapped in paper or straw pieces before packing
because paper or straw acts as buffers. It
changes the time of impact
and hence, avoids the chances of
damage during the jerks.
Ø An athlete should stop slowly, after finishing a fast race, so that the
time of impact of his run increases at stop and hence, force experienced by him
decreases.
Ø In cricket, a player lowers his
hands while catching a cricket ball to avoidinjury. In doing so, he increases
the time of impact of the ball which in turn reduces the effect of the force on
his hands.
Ø Shockers in the motor-vehicles reduce the effect of jerk/force by
increasing the time of impact of the erks given by an uneven road.
Ø In a head-on collision between two vehicles, change in linear momentum
is equal to the sum of the linear momenta of the two vehicles. Since time
impact is very small, hence an extra large force develops which results in
maximum damage to the vehicles.
Ø When a person falls from a height on a concrete floor, the floor does
not yield. The total change in linear-momentum is produced in a very small
interval of time. Hene, the floor exerts a much larger force and the person
receives more injury. But when a person falls on a heap of sand, the sand
yields. The same change in linear momentum is produced in a much longer time.
The average force exerted on the person by the heap of sand is, therefore, much
smaller and hence the person is not hurt.
Step 1 : Decide the System
The first step is to decide the system on which the laws of motion are to
be applied. The system may be a single particle, a block, a combination of two
blocks one kept over the other, two blocks connected by a string, a piece of
string etc. The only restriction is that all parts of the system should have
identical acceleration.
Consider the situation shown in figure (2).
The block B does not slip over A, the disc D slides over the string
and all parts of the string are tight.
Figure 2.
A and B move together. C is not in contact with A or B.
But as the length of the string between A and C does not change, the
distance moved by C in any time interval is same as that by A. The same
is true for G. The distance moved by G in any time interval is
same as that by A, B or C. The direction of motion is also the
same for A, B, C and G. They have identical accelerations. We can
take any of these blocks as a system or any combination of the blocks from
these as a system. Some of the examples are (A), (B), (A + B), (B + C),
(A + B + C), (C + G), (A + C + G), (A + B + C + G) etc. The
distance covered by E is also the same as the distance covered by G but
their directions are different. E moves in a vertical line whereas G in
a horizontal line. (E + G) should not be taken as a system. At least at
this stage we are unable to apply Newton's law treating E + G as a
single particle. As the disc D slides over the string the distance covered
by D is not equal to that by E in the same time interval. We
should not treat D + E as a system. Think carefully.
Step 2 : Identify the Forces
Once the system is decided, make a list of the forces acting on the
system due to all the objects other than the system. 'Any force applied by the
system should not be included in the list of the forces.
Consider the situation shown in figure (5.5). The boy stands on the
floor balancing a heavy load on his head. The load presses the boy, the boy
pushes the load upward the boy presses the floor downward, the floor pushes the
boy upward, the earth attracts the load downward, the load attracts the earth
upward, the boy attracts the earth upward and the earth attracts the boy downward.
There are many forces operating in this world. Which of these forces should we
include in the list of forces ?
Figure 3.
We cannot answer this question. Not because we do not know, but because
we have not yet specified the system. Which is the body under consideration ? Do
not try to identify forces before you have decided the system. Suppose we concentrate
on the state of motion of the boy. We should then concentrate on the forces
acting on the boy. The forces are listed in the upper half of table (5.1).
Instead, if we take the load as the system and discuss the equilibrium of the load,
Newton's Laws of Motion 67 the list of the forces will be different. These
forces appear in the lower half of table (4).
Table 4.
One may furnish as much information as one hasabout the magnitude and
direction of the forces. The contact forces may have directions other than
normal to the contact surface if the surfaces are rough. We shall discuss more
about it under the heading of friction.
Step 3 : Make a Free Body Diagram
Now, represent the system by a point in a separate diagram and draw vectors
representing the forces acting on the system with this point as the common origin.
The forces may lie along a line, may be distributed in a plane (coplanar) or
may be distributed in the space (non-planar). We shall rarely encounter situations
dealing with non-planar forces. For coplanar forces the plane of diagram
represents the plane of the forces acting on the system. Indicate the
magnitudes and directions of the forces in this diagram. This is called a free
body diagram. The free body diagram for the example discussed above with
the boy as the system and with the load as the system are shown in figure
(5)
Figure 5
Step 4 : Choose Axes and Write Equations
Any three mutually perpendicular directions may be chosen as the X-Y-Z
axes. We give below some suggestions for choosing the axes to solve
problems.
If the forces are coplanar, only two axes, say X and Y, taken in
the plane of forces are needed. Choose the X-axis along the direction in which
the system is known to have or is likely to have the acceleration. A direction
perpendicular to it may be chosen as the Y-axis. If the system is in
equilibrium, any mutually perpendicular directions in the plane of the diagram may
be chosen as the axes. Write the components of all the forces along the X-axis
and equate their sum to the product of the mass of the system and its acceleration.
This gives you one equation. Write the components of the forces along the
Y-axis and equate the sum to zero. This gives you another equation. If the
forces are collinear, this second equation is not needed.
If necessary you can go to step
1, choose another object as the system, repeat steps 2, 3 and 4 to get more
equations. These are called equations of motion. Use mathematical techniques to
get the unknown
quantities out of these equations. This completes the algorithm.
The magnitudes of acceleration of different objects in a given situation
are often related through kinematics. This should be properly foreseen and used
together with the equations of motion. For example in figure
(2) the accelerations of C and E have same magnitudes. Equations
of motion for C and for E should use the same variable a
for acceleration.
Third Law
of Motion
‘’To every action, there is always, an equal and opposite reaction.’’
Here, the action is the force exerted by one body on the other body
while the reaction is the force exerted by the second body on the first.
Significance of Third Law : It signifies that
forces in nature are always in pairs. A single isolated force is not possible. Force
of action and reaction act always on different bodies. They never cancel each
other and each force produces its own effect. The forces of action and reaction
may be due to actual physical contact of the two bodies or even from a
distance. But they are always equal and opposite. This third law of motion is
applicable whether the bodies are at rest or they are in motion. This law is
applied to all types of forces e.g. gravitational,
electric or magnetic forces, etc.
The block shown in figure (6) has a mass
M and descends with an acceleration a. The mass of the string below the point A
is m. Find the tension of the string at the point A and at the lower end.
figure (6)
Solution :
Consider "the block + the part of the string below A" as the
system. Let the tension at A be T. The forces acting on this system are
(a) (M + m)g, downward, by the earth
(b) T, upward, by the upper part of the string.
The first is gravitational and the second is electromagnetic. We do not
have to write the force by the string on the block. This electromagnetic force
is by one part of the system on the other part. Only the forces acting on the
system by the objects other than the
system are to be included. The system is descending with an acceleration
a. Taking the downward direction as the X-axis, the total force along the
X-axis is (M + m)g — T. Using Newton's law
(M+ m)g — T = (M + m)a.
Or,
T = (M + m)(g — a).
We have omitted the free body diagram. This you can do if you can draw
the free body diagram in your mind and write the equations correctly.
To get the tension T' at
the lower end we can put m = 0 in (i).
Effectively, we take the point A at the lower end. Thus, we get T'
=M(g — a).
Suppose the string in Example 5.3 or 5.4 is
very light so that we can neglect the mass of the string. Then T'= T. The
tension is then the same throughout the string. This result is of general
nature. The tension at all the points in a string or a spring is the same provided
it is assumed massless and no massive particle or body is connected in between.
If the string in figure (7) is light, the
tension T1 of the string is same at all the points between the block A and the pulley
B. The tension T, is same at all the points between
the pulley B and the block C. The tension T3 is same
at all the points between the block C and the block D. The three
tensions T1, T2 and T3 may be different from each other.
If the pulley B is also light, then T1 = T2.
figure (7)
Example and application of the third law of motion :
Ø A book placed on a table exerts a force as an action on the table. This
action is equal to the weight of the book. The table exerts a force of reaction
equal and opposite to the reaction to support the book.
Ø When a gun fires a bullet, it moves forward due to a force exerted by
the gun. The bullet exerts a reaction due to which the gun recoils backward.
Ø We can walk on a ground easily if it is tough because the ground provides sufficient reaction against our
push. But it is difficult to walk
on sand or ice. This is because on pushing, sand gets displaced
and reaction from sandy ground is very little. In case of ice,
force of reaction is again small, because friction between our feet and ice is very little.
Ø When a rubber ball is struck
against a wall or floor, it exerts a force as an action on the wall. The ball rebounds
with an equa and opposite force as reaction exerted by the
wall on the ball.
Ø A swimmer pushes the water with a
force of action in backward direction while water pushes the swimmer with a
force of reaction in the forward direction. Consequently, the
swimmer is able to swim.
Ø When a jet-plane or rocket moves
in the sky, the gases produced due to
combustion of fuel escape through the nozzle in the backward direction due to
the force of action exerted by the engine. The escaping gases exert a force of
reaction on the jet-plane or rocket in the forward direction. Consequently, the
jet-plane or rocket moves.
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