Wednesday, November 28, 2012

Amendment in Newton's second law of motion

­­­­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

{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}

where ‘p’ {\displaystyle {\vec {p}}}is the momentum of the system and ‘F’ {\displaystyle {\vec {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

{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}

{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\frac {\mathrm {d} \left(m{\vec {v}}\right)}{\mathrm {d} t}},}where m is the mass and {\displaystyle {\vec {v}}}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{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}

 

{\displaystyle {\vec {F}}=m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}.}By substituting the definition of acceleration, the algebraic version of Newton's Second Law is derived:

‘F = ma’ [16] [17] [18]{\displaystyle {\vec {F}}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}},}

 

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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