Mathematics
Newton's Second Law
Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This law is expressed by the equation F = ma, where F represents force, m is mass, and a is acceleration. It provides a quantitative relationship between the force applied to an object and the resulting acceleration.
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10 Key excerpts on "Newton's Second Law"
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
4.2.2 N EWTON ’ S S ECOND L AW The proper mathematical statement of Newton’s second law takes the following form: F p net t = ∆ ∆ , (4.1) where p is the momentum of the object defined as p = m v , (4.2) m is its mass. From the above two equations one may have F v net m t = ∆ ∆ ( ) . Assuming that the mass stays constant, this equation reduces to F v net = m t ∆ ∆ , or F = m a . (4.3) This is a vector equation equivalent in two dimensions to two-component equations: (F x ) net = ma x , (4.4a) (F y ) net = ma y . (4.4b) EXAMPLE 4.1 Calculate the horizontal force with which a 6-year-old child is pulling her 1.40-kg toy box on a smooth surface so that it moves in a straight line with an acceleration of 1.20 m/s 2 . S OLUT ION Since the motion is one dimensional, say x, the acceleration is then along the x direction. Thus, Newton’s second law, (F x ) net = ma x , which after substitution becomes (F x ) net = (1.40 kg)(1.20 m/s 2 ) = 1.68 N. 67 Newton’s Laws: Implications and Applications © 2010 Taylor & Francis Group, LLC A NALYSIS Such a force is moderately small with which, as could be envisioned, a 6-year-old child can move such mass. EXAMPLE 4.2 How much force is needed to give a truck of 7.00 × 10 3 kg an acceleration of 5.00 m/s 2 ? S OLUT ION Using Newton’s second law, (F x ) net = ma x , the force would then be (F x ) net = (7.00 × 10 3 kg)(5.00 m/s 2 ) = 3.50 × 104 N. A NALYSIS This is a large force that is needed to move a 7.00-ton truck. 4.2.3 N EWTON ’ S T HIRD L AW Newton’s third law is a fundamental law that describes an essential feature of nature. It describes mutual physical forces between any pair of entities when one of them acts with a force on the other. The object that is being acted on by a force responds, that is, reacts, instantaneously with a force of reaction equal and opposite in direction to the force acting on it.
eBook - ePub- Hiqmet Kamberaj(Author)
- 2021(Publication Date)
- De Gruyter(Publisher)
These examples indicate that the acceleration of an object is directly proportional to the resultant force acting on it. The acceleration of an object should also depend on its mass. For example, consider the following experiment: If you apply a force F to a block of ice moving on a horizontal frictionless surface, then the block undergoes some acceleration, a. If you double the mass of the block, then the same applied force produces an acceleration of a / 2. If the mass is tripled, then the same applied force delivers acceleration of a / 3, and so on. This observation indicates that the magnitude of the acceleration of an object is inversely proportional to its mass. These observations lead to Newton’s second law. Newton’s second law The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. We can relate mass and force using the following expression, which represents a mathematical statement of the second law of the Newton: (5.7) ∑ i F i = m a. Since this expression is a vectorial equation, we can project along the three coordinate axes as follows: (5.8) ∑ i F x i = m a x (5.9) ∑ i F y i = m a y (5.10) ∑ i F z i = m a z. In the SI unit system, the force has the units of newton (N), which is defined as the force acting on a mass of 1 kg gaining an acceleration. of 1 m/s 2. From the definition of Newton’s second law, we see that the newton can be expressed in terms of the units of mass, length, and time as follows: (5.11) 1 N ≡ 1 kg · m s 2. In the engineering system or British system, the unit of force is pound, which is defined as the force acting on a mass of 1 slug 1 to produce an acceleration of 1 ft/s 2 : (5.12) 1 lb ≡ 1 slug · ft/s 2. An approximation is (5.13) 1 lb ≈ 1 4 N. 5.5 Newton’s third law When we press against a corner of a textbook with the fingertip, the book pushes back and makes a small dent in our skin
eBook - PDFApplied Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
For example, if the force is applied for 10 seconds, the change of momentum will be double that achieved by applying the same force for only 5 seconds. Consequently, the second law is sometimes stated as: The change of momentum is directly proportional to the applied force and the time for which it acts. To obtain a more concise statement of the relation between momentum and force, let us suppose that the application of the force lasts for time /, and that during this time the velocity of the mass m changes from u to ν in the same straight line as the force. Now the change in momentum is mv — mu =m(v — u) in time t. Therefore the rate of change of momentum is —— per unit of time. If m is measured in kilogrammes, ν and u in metres per second, and / in seconds, then the rate of change of momentum is in kilo-gramme metres per second per second (abbreviated to kg m s 2 ). But, from the equations of motion discussed in Chapter Four, we know that ν = u + at where a is the acceleration of the body concerned, and from this equation ν — u Using this result we now see that hence the rate of change of momentum is ma kg m s 2 . Newton's Second Law enables us to say that the applied force F is directly proportional to ma, which means that we may write where b is a constant. F = o m a At this stage we realize we have yet to define the unit of force; we made use of the unit of force in Chapters One and Two solely for descriptive purposes. Clearly it would be an advantage if we could arrange for b in the equation to have a numerical value of 1, so that F = ma. With the standard units beingl kg for m and 1 m s 2 for a, this means the standard unit for F is 1 kg m s ~ 2 . There-fore the unit force is that force which will give a mass of 1 kg an acceleration of l m s -2 . We call this unit of force a newton and use Ν for its abbreviation.
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
In fact, physicists have carried out very sen-sitive tests of this hypothesis, without finding any variation. So, we can treat mass as a scalar, i.e. a simple number, and write F = m a . This is Newton’s second law of motion, which will underlie much of our subsequent discussion. It is worth emphasizing that force is not merely a matter of definition. For instance, if we observe that an air track rider of mass m starts to accelerate at rate a , it might be tempting to conclude that we have just observed a force F = m a . Tempting, but wrong. The reason is that forces always arise from real physical interactions between systems. Interac-tions are scientifically significant: accelerations are merely their conse-quence. Consequently, if we eliminate all interactions by isolating a body su ffi ciently from its surroundings—an inertial system—we expect it to move uniformly. You might question whether it is really possible to totally isolate a body from its surroundings. Fortunately, as far as we know, the answer is yes . Because interactions decrease with distance, all that is required to make interactions negligible is to move everything else far away. The forces that extend over the greatest distance are the familiar gravita-tional and Coulomb electric forces. These decrease as 1 / r 2 , where r is the distance. Most forces decrease much more rapidly. For example, the force between separated atoms decreases as 1 / r 7 . By moving bodies suf-ficiently far apart, the interactions can be reduced as much as desired. 2.6 Newton’s Third Law That force is necessarily the result of an interaction is made explicit by Newton’s third law. The third law states that forces always appear in 2.6 NEWTON’S THIRD LAW 55 pairs that are equal in magnitude and opposite in direction: if body b exerts force F a on body a , then there must be a force F b acting on body b , due to body a , such that F b = − F a .
eBook - ePubNewtonian Dynamics
An Introduction
- Richard Fitzpatrick(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
Nowadays, Newton’s first law strikes us as almost a statement of the obvious. However, in Galileo’s time, this was far from being the case. From the time of the ancient Greeks, philosophers—observing that objects set into motion on the Earth’s surface eventually come to rest—had concluded that the natural state of motion of objects was that they should remain at rest. Hence, they reasoned, any object that moves does so under the influence of an external influence, or force, exerted upon it by some other object in the universe. It took the genius of Galileo to realize that an object set into motion on the Earth’s surface eventually comes to rest under the influence of frictional forces, and that, if these forces could somehow be abstracted from the motion then the motion would continue forever.4.3 Newton’s Second Law of Motion
Newton used the word “motion” to mean what we nowadays call momentum. The momentum, p, of a body is defined as the product of its mass, m, and its velocity, v; that is,Newton’s second law of motion is summed up in the equationp = m v .(4.2)(4.3)= f ,d pd twhere the vector f represents the net influence, or force, exerted on the object, whose motion is under investigation by other objects in the universe. For the case of a object with constant mass, the previous law reduces to its more conventional formm a = f .(4.4)In other words, the product of a given object’s mass and its acceleration,a = d v / d t, is equal to the net force exerted on that object by the other objects in the universe. Of course, this law is entirely devoid of content unless we have some independent means of quantifying the forces exerted between different objects.4.4 Measurement of Force
One method of quantifying the force exerted on an object is via Hooke’s law. (See Section 5.6 .) This law—discovered by the English scientist Robert Hooke in 1660—states that the force, f, exerted by a coiled spring is directly proportional to its extension,Δ x
eBook - PDFApplied Mechanics
Made Simple
- George E. Drabble(Author)
- 2013(Publication Date)
- Made Simple(Publisher)
The first law gives us a definition of force: force is something which, by itself, produces an acceleration. This is the definition of force, and is clearly more satisfactory than the 'push or pull' we have had to accept so far. It only gives us a qualitative definition: it does not tell us how to measure force, but we shall find the answer to this problem in the second law. Let us now see how general the first law is in its application, and how it must have appeared to cut right across contemporary beliefs. It is natural to assume that a body which is not acted upon by a force would be in a state of rest: it is not so obvious to assume a possible state of motion in a straight line. All terrestrial experience of Newton's day must have pointed to other con-clusions. All bodies on the Earth, if set moving, came 'naturally' to rest. On the other hand, bodies apparently free from earthly interference (planets, for instance) were known to move in approximately circular paths. From the time of the Greeks, one school of thought accepted the 'natural' motion of bodies as circular. We now know that earth-bound bodies are subjected to fric-tional force, and that planets are subjected to gravitational force: both of 40 Applied Mechanics Made Simple these forces cause departure from straight-line uniform motion. In fact, perhaps the most exasperating aspect of Newton's theories is that no body ever observed has a 'natural' motion unaffected by force. There is an intri-guing and amusing dialogue on this topic between Newton and the artist, Kneller in G. B. Shaw's play, 'In Good King Charles's Golden Days'. It is perhaps typical of Newton's genius that he was able to perceive, without the benefit of direct observation, what the unforced motion of a body would be. Present-day students have less difficulty with this idea than those of an earlier age.
No longer available |Learn moreRIGHTS REVERTED - Reasoning About Luck
Probability and Its Uses in Physics
- Vinay Ambegaokar(Author)
- 2017(Publication Date)
- Dover Publications(Publisher)
Suppose that an object is moving steadily around a circle of radius one meter at such a rate that it makes one circuit in 2 π seconds, i.e. approximately 6.28 seconds. Then, using (6.6), we find that its acceleration towards the center is one mks unit – one meter/(second) 2. If the object has a mass of one kilogram, Newton’s second law then informs us that this requires a steady force pulling it towards the center of one mks unit of force, or one newton. The one other situation I would like to explain is easier to understand : constant acceleration in the direction of the velocity. This is, approximately, what happens when an object is dropped from a window. [Approximately because the gravitational force is not the only force acting on such a falling object; there is air resistance as well.] The force of gravity is proportional to the mass of the object; this force causes all bodies falling near the surface of the earth to accelerate downward at the same rate, which in mks units is approximately 9.81 m/ s 2. An object released from rest thus picks up downward velocity as it accelerates. The ubiquity of frictional forces opposing motions in our everyday world makes it easy to understand why astronomical observations played such an important role in the insights of Newton and his contemporaries. On the other hand, although the original impetus for the development of mechanics may have been the explanation of celestial motions, one of Newton’s great triumphs was the unification of celestial and terrestrial dynamics. We are very close to being able to understand this synthesis. What follows is a small digression in that direction
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
The object experiences acceleration due to gravity. • Some upward resistance force from the air acts on all falling objects on Earth, so they can never truly be in free fall. Chapter 5 | Newton's Laws of Motion 253 • Careful distinctions must be made between free fall and weightlessness using the definition of weight as force due to gravity acting on an object of a certain mass. 5.5 Newton’s Third Law • Newton’s third law of motion represents a basic symmetry in nature, with an experienced force equal in magnitude and opposite in direction to an exerted force. • Two equal and opposite forces do not cancel because they act on different systems. • Action-reaction pairs include a swimmer pushing off a wall, helicopters creating lift by pushing air down, and an octopus propelling itself forward by ejecting water from its body. Rockets, airplanes, and cars are pushed forward by a thrust reaction force. • Choosing a system is an important analytical step in understanding the physics of a problem and solving it. 5.6 Common Forces • When an object rests on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force. • When an object rests on a nonaccelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object. • When an object rests on an inclined plane that makes an angle θ with the horizontal surface, the weight of the object can be resolved into components that act perpendicular and parallel to the surface of the plane. • The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension. When a rope supports the weight of an object at rest, the tension in the rope is equal to the weight of the object. If the object is accelerating, tension is greater than weight, and if it is decelerating, tension is less than weight.
- Allan D. Kraus, James R. Welty, Abdul Aziz(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
14 Newton’s Second Law of Motion Chapter Objectives • To introduce the control volume relationship for linear momentum. • To provide applications of the momentum theorem. • To consider the control volume relationship for the moment of momentum. • To use applications taken from a number of physical situations to show applications of the moment of momentum relationship. 14.1 Introduction The third fundamental law to be considered is Newton’s second law . As in Chapter 13, we will develop control volume expressions, which, in this case, will be related to both linear and angular motion. The basic expressions will then be applied to a number of physical situations. 14.2 Linear Momentum Newton’s second law of motion may be stated as The time rate of change of momentum of a system is equal to the net force on the system and takes place in the direction of the net force. This statement is notable in two ways. First, it is cast in a form that includes both magni-tude and direction and is, therefore, a vector expression. Second, it refers to a system rather than a control volume. As we know by now, a system is a fixed collection of mass, whereas a control volume is a fixed region in space that encloses a different mass of fluid (or system) at different times. The transformation of the second law statement from a system to a control volume point of view is dealt with in numerous texts. We will presume the correctness of the following word equation: ⎧ ⎨ ⎩ Net force acting on the control volume ⎫ ⎬ ⎭ = ⎧ ⎨ ⎩ Rate of momentum out of the control volume by mass flow ⎫ ⎬ ⎭ − ⎧ ⎨ ⎩ Rate of momentum into the control volume by mass flow ⎫ ⎬ ⎭ + ⎧ ⎨ ⎩ Rate of accumulation of momentum within the control volume ⎫ ⎬ ⎭ (14.1) 441 442 Introduction to Thermal and Fluid Engineering m e Control Volume m i . . FIGURE 14.1 A general control volume and flow field. The control volume shown in Figure 14.1 has the same general features that are shown in Figure 13.1.
eBook - ePub- Charles Liu(Author)
- 2020(Publication Date)
- Visible Ink Press(Publisher)
How can these two quantities be equal? The answer can be found by looking at Newton’s Second Law again. In the form F net = ma, you can see that the units of force, newtons, must be equal to the units in which m times a are measured. The mass, m, is measured in kilograms and the acceleration in meters per second squared. Therefore, a newton must be a kilogram times a meter per second squared. Thus, a newton per kilogram (N/kg) is a meter per second squared (m/s 2). How do the speed and position of a dropped object vary with time? As long as the only force is the gravitational force, then the acceleration is the acceleration due to gravity, g. At Earth’s surface the value is 9.8 m/s 2. (In the English system, g = 32.2 ft/s 2 or about 22 mph per second.) If the object is dropped from rest at time t = 0, then the velocity at a future time t is simply v = gt. That is, the velocity increases by 9.8 m/s each second. If we measure the distance fallen from the position where it was dropped, then at time t, it has fallen a distance d = 1/2 gt 2. The following table shows velocity and distance fallen for some selected times. How can I use a table of velocity and distance fallen to conduct experiments? The first three lines on the table above have been chosen so that you can explore your reaction time—the amount of time it takes you to move your hand once your eye has seen something. To conduct this experiment, have another person hold a ruler or yardstick vertically by one end. Place your finger and thumb at the lower end of the ruler, but don’t let them touch the ruler. Have the other person drop the ruler, and you use your finger and thumb to grab it. Note the distance the ruler has dropped in the time it takes you to react to it being dropped
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