EFFECT OF VERTICAL
FORCES ON NEWTON’S SECOND LAW OF MOTION
ABSTRACT
This paper
proposed a more general expression to Newton’s second law of motion (F=ma) to
calculate the force acting on the bodies under various factors that affects net
force. As more force is required to move an object having larger mass but what
happen when weight is changed by keeping its mass same. According to Newton’s
second law, value of force varies, even if we apply same force. This happens
because if force is not applied in vacuum, other forces add up to the equation ‘F=ma’ to get net force required to move
the body. It is difficult to calculate other forces and add up to the given
convention. This paper explains, weight includes the variation of all vertical forces.
Hence, the
simplest suggestive expression is ‘F=wa’
to study the horizontal motion of an object under vertical forces.
Keywords: Newton second law, effect of weight, vertical
forces, mass, F=wa, F=ma.
INTRODUCTION
Philosophers in antiquity
used the concept of force in the study of stationary and moving objects, but thinkers
such as Aristotle retained fundamental errors in understanding this concept of
force due to an incomplete understanding of non-obvious force of friction. It
was belief that a force is required to maintain motion, even at a constant
velocity but actually this force is only required to stop its motion, not to
move it with constant velocity. Galileo Galilei and Sir Isaac Newton eventually
corrected most of the previous misunderstandings about motion and force. With
some mathematical insight, Sir Isaac Newton formulated laws of motion that were
not improved-on for nearly three hundred years. In many ways Newton’s work was
a revolutionary departure from previous ideas about motion, it owed an immense
debt to Galileo’s exhaustive study of motion. Newton and Galileo saw, as others
had not, what motion would be like in a vacuum.
At the first, the laws of motion were
proposed by Aristotle (384-322 B.C.). In observing objects around him,
Aristotle noticed that an object required a force in order to start moving. If
that force was removed, the object eventually came to stop. A large force
resulted in a large velocity; a small force resulted in a small velocity. He
concluded that force was related to velocity. In some situations, however,
force resulted in no velocity. We could push and push on a heavy boulder, but
it would never budge. In order to explain these observations, Aristotle added
the concept of resistance. Resistance to motion can arise from two sources. If
the resistance is offered by the object we are trying to move, then resistance
is a property of that object – like the later ideas of mass and inertia. If the
resistance is offered by the material through which the object moves, then
resistance is like friction – a force exerted by other materials. Aristotle
chose the latter, calling resistance a property of the medium through which an
object moves. Combining the concept of resistance with the idea that force was related
to velocity, Aristotle formulated his “second law;”
Velocity = force /
resistance
[1] [2] [3] [4]
It looks remarkably like Newton’s second
law, except that it relates force to velocity instead of acceleration, and it
defines resistance as a property of the medium, not the object being moved.
When Aristotle tried to imagine what would happen in a vacuum, he saw an
absurdity. In a vacuum, the resistance would be zero and the objects velocity
would increase to infinity. Aristotle concluded that vacuums did not exist.
In the years from Aristotle to Galileo,
Philoponus, Avicenna, and a number of other scientists grappled unsuccessfully
with this problem of motion in a vacuum. John Philoponus (490-570AD)) attacked this problem of motion in a
vacuum by suggesting that Aristotle’s relationship be modified as the speed
(motion) is determined by the excess of the force taken as a difference to the
resistance or
Velocity = force –
resistance
[1] [2] [3] [4]
If the resistance were zero, the object
would move with a constant velocity directly proportional to the force.
Consequently, motion in a vacuum would be possible. By suggesting that an
object’s motion depends on the difference between force and resistance,
Philoponus modification introduced what we have come to call the net force.
However, Philoponus still related force to velocity, not acceleration.
Once
projectile motion was understood in terms of an impetus in this way, it became
possible for Philoponus to reassess the role of the medium: far from being
responsible for the continuation of a projectile’s motion it is in fact an
impediment to it. On this basis Philoponus concludes, against Aristotle, that
there is in fact nothing to prevent one from imagining motion taking place
through a void. As regards the natural motion of bodies falling through a
medium, it was Aristotle’s contention that the speed is proportional to the
weight of the moving bodies and indirectly proportional to the density of the
medium. Philoponus repudiates this view by appeal to the same kind of
experiment that Galileo was to carry out centuries later. [5]
Avicenna (A.D. 980-1037) proposed a
different modification of Aristotle’s relationship. Avicenna suggested that
objects themselves have a property, which he called mail, that resists a change
in motion. Objects moving in a vacuum would continue moving forever, not speed
up forever as Aristotle had supposed. Since Avicenna had never seen objects
that moved at a constant velocity forever, he, too, concluded that vacuums do
not exist. Like Aristotle, Avicenna failed to pursue the question of what would
happen if. [4] He developed
a theory of motion in ‘The Book of Healing’. In this book, he made a distinction
between the inclination (tendency to motion) and force of a projectile, and concluded
that motion was a result of an inclination transferred to the projectile by the
thrower, and that projectile motion in a vacuum would
not cease. He viewed inclination as
a permanent force whose effect is dissipated by external forces such as air resistance. The theory of
motion presented by Avicenna was probably influenced by the 6th-century
Alexandrian scholar John Philoponus. His theory of motion was thus consistent with the
concept of inertia in Newton's first law of motion. Avicenna also referred to
as being proportional to weight times velocity and that such a moving body
would eventually be slowed down by the air resistance around it. [6] [7]
[8]
John Buridan’s (1301–1359/1361) major contribution here was to
develop the theory of impetus to explain projectile motion. The theory of
impetus probably did not originate with Buridan, but his account appears to be unique
in that he entertains the possibility that it might not be self-dissipating:
“after leaving the arm of the thrower, the projectile would be moved by an
impetus given to it by the thrower, and would continue to be moved as long as
the impetus remained stronger than the resistance, and would be of infinite
duration were it not diminished and corrupted by a contrary force resisting it
or by something inclining it to a contrary motion”. He also contends that
impetus is a variable quality whose force is determined by the speed and
quantity of the matter in the subject, so that the acceleration of a falling
body can be understood in terms of its gradual accumulation of units of
impetus. [11] The
implanted impetus is caused by a mover who imparts an initial velocity to a
projectile; the impetus is proportional to the velocity: in fact, Buridan
gave it a mathematical formulation:
Impetus = weight ×
velocity [8] [9] [10] [11]
When a mover sets
body in motion, he implants into it a certain impetus, that is, a
certain force enabling a body to move in the direction in which the mover
starts it, be it upwards, downwards, side wards, or in a circle. The
implanted impetus increases in the same ratio as the velocity. It is
because of this impetus that a stone move on after the thrower has ceased
moving it but because of the resistance of the air (and also because of the
gravity of the stone) which strives to move it in the opposite direction to the
motion caused by the impetus, the latter will weaken all the time. Therefore,
the motion of the stone will be gradually slower, and finally the impetus is so
diminished or destroyed that the gravity of the stone prevails and moves the
stone towards its natural place. [12]
Galileo (1564-1642) eventually made the
mental jump from the observed to the hypothetical – from motion in everyday
experience to motion without resistance. As shown in the figure-1
Figure-1
Galileo Experiment
As going down an incline, he reasoned, a
ball accelerates. If he placed a second incline facing the first, the ball
would move up the second incline almost to the same height from which it had
been released on the first incline. Mentally, Galileo removed friction and
concluded that the ball would continue up the second incline until it reached
exactly the same height from which it had been released. Next, Galileo
decreased the angle of the second incline. Each time, the ball traveled until
it almost reached the same height from which it had been released. At lower inclines,
however, the ball had to travel farther. If the angle of incline were reduced
to zero, Galileo reasoned, the ball would continue moving forever. [13]
[14] [15]
It is only the resistance offered by the
surface and the air that keeps this from being so. Called the law of inertia,
Galileo’s conclusion paved the way for Newton’s first law of motion. Because of
friction, Galileo was never able to actually observe an object moving along a
level board with an unchanging velocity. But he could mentally remove friction
and imagine what would happen. His guide in performing these imaginary
experiments was the simplicity of the mathematical relationships he had
discovered from actual measurements. A commitment to experimentation and
simplicity allowed Galileo to see what others had not seen motion in a
frictionless world. Galileo perceived experiments in hypothetical system when all
resistive forces are eliminated from the system, then body once set in motion
always remains in state of uniform velocity (constant velocity). Thus Galileo
on the basis of such experiments perceived that a moving body maintains its
constant speed in straight line unless no external force acts on it. Thus
Galileo put forth that movement of body with constant velocity is natural
tendency of body, and it stops due to resistive forces Galileo’s law is given
by
“A body moving on a level surface
will continue to move in the same direction at a constant speed unless
disturbed.” [16] [17] [18]
It stated law of inertia that body will maintain its
uniform motion forever (in hypothetical system). However, such a system is
completely hypothetical, as it has to be devoid of resistive forces. Such
system does not exist on the Earth as Gravitational forces and frictional
forces are always present.
Rene Descartes (1596 – 1650),
in 1644 in his book Principles
of Philosophy elaborated
Galileo’s law of inertia in first two laws of motion 3-4. Further, Descartes
third law of motion explains the collision of moving bodies; it is independent
of Galileo’s law of inertia or Aristotle’s assertion. But Newton’s third law of
motion has resemblance with this law.
Law
1
Each thing, in so far as it is
simple and undivided, always remains in the same state, as far as it can, and
never
changes except as a result of external
causes. Hence we must conclude that what is in motion always, so far as it can, continues to move.
Law
2
Every piece of matter, considered in
itself, always tends to continue moving, not in any oblique path but only in a straight line.
Law
3
When a moving body collides with
another, if its power of continuing in a straight line is less than the
resistance of the other body, it is deflected so that, while the quantity of
motion is retained, the direction is altered; but if its power of continuing is
greater than the resistance of the other body, it carries that body along with
it, and loses a quantity of motion equal to that which it imparts to the other
body. [19] [20] [21]
It is evident that Descartes first
two laws are just other form of Galileo’s law of inertia. Descartes third law
of motion is independent of Galileo’s Laws of inertia. Thus it is Descartes
original work and preceded Newton’s third law of motion. Neither Galileo nor
Descartes gave any hint to find out the magnitude of force required for change
the constant velocity of body.
Isaac Newton (1643-1727) built upon Galileo’s work, adding the
concepts of force and mass to Galileo’s descriptions of motion. Newton’s second
law of motion explains how the velocity of an object changes when it is
subjected to an external force. The law defines a force to be equal to change
in momentum (mass times velocity) per change in time. Newton also developed the
calculus of mathematics, and the "changes" expressed in the second
law are most accurately defined in differential forms. A modern statement of
Newton's Second Law is a vector equation:
F =
dp/dt
where ‘p’ is the momentum of the system and ‘F’ is the net force. If a body is in equilibrium, there
is zero net force by
definition (balanced forces may be present nevertheless). In contrast, the
second law states that if there is an unbalanced force acting on an object it will
result in the object's momentum changing over time. By the definition of momentum;
F =
dp/dt = d(mv)/dt
where m is the mass and v is the velocity. If Newton's
second law is applied to a system of constant
mass
m may be moved
outside the derivative operator. The equation then becomes
F = m
dv/dt
By substituting the definition of acceleration,
the algebraic version of Newton's Second Law is derived:
‘F = ma’ [16]
[17] [18]
For an external applied force, the change
in velocity depends on the mass of the object. A force will cause a change in
velocity; and likewise, a change in velocity will generate a force. The
equation works both ways. Newton's Second Law asserts the direct proportionality
of acceleration to force and the inverse proportionality of acceleration to
mass. Accelerations can be defined through kinematic measurements.
EXAMINATION OF
NEWTON’S SECOND LAW OF MOTION
It has been ages since Newton’s second law of motion
is being used in every nook and corner of the physics. It is observed that results with the
convention ‘F=ma’ doesn’t show net force as it encounters various forces on
Earth. It seems if mass and acceleration remains same but weight increases
w.r.t. vertical forces, force will have to increase. This paper considers some
analysis of the formula. There are some such circumstances.
Case-I
Let
some force is applied to a stationary object of mass 100kg under the gravity of
Pluto and Earth, i.e. weight of the same object becomes 62N on Pluto and 980N
on Earth. How much force is needed to get final velocity of 1m/s in 10sec on
both planets?
Mass = 100Kg Weight
= 62N
Mass = 100Kg Weight
= 980N
Acceleration = [(v - u) ÷ t]
a = [(1 - 0) ÷ 10]
a = 0.1m/s2
On Pluto,
Force = mass × acceleration
F = 100kg × 0.1m/s2
F = 10N
On Earth,
Force = mass × acceleration
F = 100kg × 0.1m/s2
F = 10N
In this case, the object of mass100kg
produced final velocity of 1m/s in 10 seconds when 10N force was applied to it
under different gravities (weighs 62N and 980N). It concludes that we need same
force under different gravities to move the same object with same final
physical quantities (final velocity, initial velocity and time) but in actual
practice, how is it possible? Since object weighs different under different
gravities, therefore as per its weight due to change in gravity, it actually
needs different force to move with same final physical quantities. As per our
opinion, it is the weight which actually works not only the mass because weight
includes the variation of all vertical forces such as mass and gravity but in
Newton’s law it only includes the variation of mass not the gravity. In fact,
‘F=ma’ explains that if weight increases due to mass, it needs more force but
if weight increases due to gravity it needs same force, which seems to be quite
imprecise. It seems that more force is required to move an object with same
velocity and acceleration on the surface of the Earth rather than on the Pluto.
This is because Earth has more gravity than Pluto which increases the weight.
Case-II
Let, take an iron block of mass 1kg (weight 9.8N on Earth) and place it on
the wooden table. Now, the iron block is hit with a paint ball gun from
one-meter distance with 120N force then
acceleration would be,
F = ma
120N = 1kg. a
a = 120N ÷ 1kg
a = 120m/s2
Further, take an iron block of mass 10kg (weight 98N on Earth) and place it on the
wooden table. Now, the iron block is hit with the same paint ball gun from same
distance with same force then
acceleration would be,
F = ma
120N = 10kg. a
a = 120N ÷ 10kg
a = 12m/s2
It
is clear that if same force is applied, when mass or weight increases the
acceleration decreases with the same rate.
Furthermore, take an iron block of mass
1kg (weight .62N on Pluto) and place
it on the wooden table. Now, the iron block is hit with the same paint ball gun
from same distance with same force then
acceleration would be,
F = ma
120N = 1kg. a
a = 120N ÷ 1kg
a = 120m/s2
In another scenario, take mass 1kg and magnet
is placed beneath the wooden table and let the weight of the same block becomes
98N. Now, hit it with the paint ball gun from distance 1meter (i.e. with 3N force) keeping all other physical properties same.
F = ma
120N = 1kg × a
a = 120m/s2
|
|
Weight changed
due to Earth gravity |
Weight changed
due to Earth gravity |
Weight changed
due to Pluto gravity |
Weight changed
due to magnet |
|
Applied Force |
120N |
120N |
120N |
120N |
|
Mass |
1kg |
10kg |
1kg |
1kg |
|
Weight |
9.8N |
98N |
0.62N |
98N |
|
Acceleration |
120m/s2 |
12m/s2 |
120m/s2 |
120m/s2 |
Table-1
Above observation concludes that
it doesn’t matter, weight increases hundred times or thousand due to external
forces; if same force applied to same mass, acceleration will remain same. In
actual practice, it is not possible, as various external forces such as
gravity, effect of magnet etc. act on body. Therefore, net force would be
F = ma + effect of magnet/other forces
However,
it is not easy to identify and calculate each and every vertical external force
acting on the body. Hence, the paper suggests a new equation that includes the
variation of all vertical forces when force is applied horizontally.
EQUATION OF
NEWTON’S SECOND LAW OF MOTION UNDER THE EFFECT OF VERTICLE FORCES
Finding formula of force was a puzzle for researchers and hence researchers
made time-to-time amendments. In this series, this
paper tries to explain that force encounters various external forces when
applied under gravity, i.e. force acts differently when applied horizontally under
variation of external vertical forces. It is noted that weight includes the
variation of all vertical forces. Hence, the paper suggests that
‘Second law of motion under gravity should
state that force applied on a body is directly proportional to the weight and
acceleration’
F α wa
F = k. wa
Where
‘k’ is a constant factor called the duke’s constant; includes all vertical
external forces, i.e. gravity etc. (in SI units of ‘k’ is: s2/m).
NUMERICAL
TREATMENT
This paper explains that weight changes when it
encounters vertical forces, hence a new equation is suggested that includes all
vertical forces, i.e. ‘F = k wa’. The equation explains that if mass remain
same but weight changes due to vertical forces, then acceleration varies w.r.t.
change in weight.
Let, take an iron block of mass 1kg
(weight 9.8N on Earth) and place it
on the wooden table. Now, the iron block is hit with the same paint ball gun
from same distance with same force then
acceleration would be,
F = k. wa
120N = 1s2/m × 9.8N × a
a = 120N ÷ (1s2/m × 9.8N)
a = 12.3m/s2
Further, take an iron block of mass 10kg
(weight 98N on Earth) and place it on the wooden table. Now, the iron block is
hit with the same paint ball gun from same distance with same force then acceleration would be,
F = k. wa
120 = 1s2/m × 98N × a
a = 120N ÷ (1s2/m × 98N)
a = 1.23m/s2
It
is clear that if same force is applied, when mass or weight increases the
acceleration decreases with the same rate.
Furthermore, take an iron block of mass
1kg (weight .62N on Pluto) and place
it on the wooden table. Now, the iron block is hit with the same paint ball gun
from same distance with same force then
acceleration would be,
F = k. wa
120N = 1s2/m × 0.62N × a
a = 120N ÷ (1s2/m × 0.62N)
a = 193m/s2
In another scenario, we take mass 1kg and magnet
is placed beneath the wooden table and let the weight of the same block become 98N. Now, hit it with the paint ball from same distance with same force keeping all other physical properties
then acceleration would be,
F = wa
120Ng = 1s2/m × 98N × a
a = 120Ng ÷ (1s2/m × 98N)
a = 1.23m/s2
|
|
Weight changed
due to Earth gravity |
Weight changed
due to Earth gravity |
Weight changed
due to Pluto gravity |
Weight changed
due to magnet |
|
Applied Force |
120N |
120N |
120N |
120N |
|
Mass |
1kg |
10kg |
1kg |
1kg |
|
Weight |
9.8N |
98N |
0.62N |
98N |
|
Acceleration |
12.3m/s2 |
1.23m/s2 |
193m/s2 |
1.23m/s2 |
Table-2
Here, the mass was same but
weight was increased. Hence, the acceleration is 1.23m/s2 as it
includes all vertical forces, i.e. force due to magnet force, gravity etc. It
tries to convey that weight includes all vertical forces.
EFFECT
OF ‘F=k. wa’ ON OTHER PHYSICAL QUANTITIES
I)
WORK DONE
When a force acts on a body or object, it causes the object to move
to cover a distance. Furthermore, sometimes it happens that the direction of the
moving object is not as same as the direction of the force. Moreover, in these
cases the component of force that acts in the direction of the movement causes
work to be done. In simple words, work refers
to the force you apply to an object to cause a change in its position or when
the object moves and cover some distance.
‘Net work done = m × a × S‘
When work is
done under gravity then it directly encounters various external vertical
forces. The more vertical forces, the more work will have to be done.
Therefore, net work done should be directly proportional to the weight instead
of mass only, as weight includes variation of all vertical forces. Hence
the net work done should be
For Horizontal,
Net work done
under gravity = k × weight × acceleration × Displacement
‘Workdone = k. w × a × S’
For e.g. If
an object of mass 10kg is dragged horizontally at an acceleration of 1.2m/s2
to 100m under the gravity of Earth and Pluto, then calculate work done on both
planets?
Earth,
Net Work done
against gravity = k × weight × acceleration × Displacement
W = 1s2/m × 98N × 1.2m/s2
× 100m
W = 11760Joule
Pluto,
Net Work done
against gravity = k × weight × acceleration × Displacement
W = 1s2/m × 6.2kgm/s2 × 1.2m/s2 × 100m
W =
744Joule
In
this case, it is observed that when 10kg mass is dragged horizontally at an
acceleration of 1.2m/s2 to 100m under the Earth’s gravity, then work
done will be 11760J whereas under the gravity of Pluto when same mass is
dragged to same distance at the rate of same acceleration, then the work done
will be 744J. It concludes that work done is directly affected by gravity. Here
the unit is Joule gravity but not Joule, because the work is done under
gravity.
II)
POTENTIAL ENERGY
Potential energy is energy
that is stored or
conserved in an object or substance. This stored energy is based on the
position, arrangement or state of the object or substance.
Potential Energy
= Force × Displacement
‘P.E. = mgh’
For
e.g. A body of mass 35kg is lifted to a height of 4m under the
gravity of Earth and Pluto. What is the gravitational potential energy on both
planet?
Earth,
P.E. = mgh
P.E. = 35kg ×
9.8m/s2 × 4m
P.E. = 1372J
Pluto,
P.E. = mgh
P.E. = 35kg ×
.62m/s2 × 4m
P.E. = 86.8J
In
this case, it is observed that a mass 35kg has 1372J potential energy if it is
lifted to a height of 4m under Earth’s gravity while on the other side the same
object has 86.8J potential energy when it is lifted to same height under the
gravity of the Pluto. The observations conclude gravity directly affects the
potential energy.
III)
KINETIC ENERGY
In physics, the kinetic energy (KE) of an object is the energy by
virtue of its motion. It is defined as the work needed to
accelerate a body of a given mass from rest to its stated velocity and measured as
‘K.E. = ½ mv2’
The above
equation of Kinetic energy should be used only when an object is moving without
gravity but when an object moves horizontally under gravity, it directly
encounter gravity so the equation should be,
‘K.E. = ½ wv2’
For
e.g. A ball of mass 1kg is thrown so that it has a velocity of
11m/s under the gravity of the Earth and the Pluto. What is the kinetic energy?
Earth,
K.E. = k. ½ wv2
K.E. = 1s2/m × ½ × 9.8N × (11m/s)2
K.E. = 1s2/m × ½ × 9.8N × 121m2/s2
K.E. = 592.9J
Pluto,
K.E. = ½ wv2
K.E. = 1s2/m × ½ × 1.6N × (11m/s)2
K.E. = 1s2/m × ½ × 1.6N × 121m2/s2
K.E. = 96.8Joule
In
this case, it is observed that when a mass 1kg thrown with a velocity of 11m/s
then it generates 592.9J kinetic
energy on Earth whereas when same object thrown at same velocity on Pluto
(changed gravity) it will generate 96.8J of kinetic energy. Therefore, it clears that gravity
directly affects the kinetic energy.
CONCLUSION
It is observed that when a body encounters external forces then the net force equals to the
addition of ‘F=ma’ and external forces. There we have to calculate external
forces to get net force, which makes it complicated to calculate it. This study
explains that weight includes the variation of all vertical forces, i.e. net
force should be equal to the product of weight and acceleration which also
holds present Newton’s second law.
ACKNOWLEDGEMENTS
The Author, Amritpal Singh Nafria, would
like to thank Dr. Amit Bansal for his useful discussion during this research
work.
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