Application of Newton's Second Law

10 Key excerpts on "Application of Newton's Second Law"

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  eBook - PDF

  Essential Physics

  • 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.

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  Physics, Student Study Guide

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Friction The force that an object encounters when it moves or attempts to move along a surface. It is always directed parallel to the surface in question. Tension The tendency of a rope (or similar object) to be pulled apart due to the forces that are applied at either end. Equilibrium The state an object is in if it has zero acceleration. Mathematically, equilibrium means = 0. Apparent Weight The force that an object exerts on the platform of a scale. It may be larger or smaller than the true weight, depending on the acceleration of the object and the scale. Chapter 4 39 Newton's laws of Motion First Law An object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force. By "net" force we mean the vector sum of all of the forces acting on an object. For example, consider a spaceship in deep space, isolated from any other object or force. If the ship is stationary, it will remain so. But if the ship is moving (its rocket engines are shut down) it will continue to move in a straight line with a constant speed. So if the ship were traveling into deep space at say, 100 000 milh, it would continue to move at this speed in a straight line, even without the rocket engines firing, until an outside force acted to stop or change its motion. Second Law When a net force 2:F acts on an object of mass m, the acceleration a that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force. This statement is usually written as LF=ma or a=LF/m (4.1) The symbol 2:F represents the net force, that is, the vector sum of all the forces acting on an object. This means that the components of the forces must be examined. For example, in two dimensions, equation (4.1) becomes Equations (4.1) and (4.2) can be used to determine the units of force.

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  Applied Mathematics

  Made Simple

  • Patrick Murphy(Author)
  • 2014(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  When a body slides across a surface there is always a force of friction which opposes the continuation of the motion. We attempt to minimize this frictional force by moving the body across what are called 'smoother' surfaces, i.e. surfaces which offer little resistance to the motion across them. Ice is a particularly good surface to attempt an approximation to the first law since its resistance to motion is so small. A puck used in ice hockey will travel a considerable distance across ice, and noticeably it travels in a straight line. A steel ball-bearing will travel even farther, and again in a straight line. Each time we lessen the resistance to the motion, the closer we get to a continuation of the state of uniform motion in a straight line. We infer, therefore, that if we could only remove all resistance to the motion, i.e. have a zero-resisting force, Newton's First Law would be verified. (3) The Second Law The original statement of Newton's Second Law referred to a 'change in motion', but Newton interpreted this as being dependent on the mass of the body and its velocity. We measure the change in 'motion' by the change in momentum, so the second law is often stated as: The rate of change of momentum is directly proportional to the applied force and takes place in the direction of this applied force. Newton's Laws of Motion 69 If we look further into the influence of this applied force, we realize that the change of momentum will also depend on the length of time the force is applied. 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.

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  Applied 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.

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  Physics, Volume 1

  • Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  From this extremely precise experiment, we conclude that down to accelerations of about 10 10 m/s 2 , force is proportional to acceleration and Newton’s second law holds. In the 20th century, we have experienced three other revolutionary developments: Einstein’s special theory of relativity (1905), his general theory of relativity (1915), and quantum mechanics (in about 1925). Special relativity teaches us that we cannot extrapolate the use of Newton’s laws to particles moving at speeds comparable to the speed of light. General relativity shows that we cannot use New- ton’s laws in the vicinity of extremely massive objects. Quantum mechanics teaches us that we cannot extrapolate Newton’s laws to objects as small as atoms. Special relativity, which involves a distinctly non-New- tonian view of space and time, can be applied under all cir- cumstances, at both high speeds and low speeds. In the limit of low speeds, it can be shown that the dynamics of special relativity reduces directly to Newton’s laws. Simi- larly, general relativity can be applied to weak as well as strong gravitational forces, but its equations reduce to New- ton’s laws for weak forces. Quantum mechanics can be ap- plied to individual atoms, where a certain randomness of behavior is predicted, or to ordinary objects containing a huge number of atoms, in which case the randomness aver- ages out to give Newton’s laws once again. Within the past two decades, another apparently revolu- tionary development has emerged. This new development concerns mechanical systems whose behavior is described as chaotic. One of the hallmarks of Newton’s laws is their ability to predict the future behavior of a system, if we know the forces that act and the initial motion. For exam- ple, from the initial position and velocity of a space probe that experiences known gravitational forces from the Sun and the planets, we can calculate its exact trajectory.

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  University Physics Volume 1

  • 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.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Since many car collisions occur at speeds much larger than 5 mi/h, The Convincer demonstrates that seatbelt use is essential for passenger safety. Concept Summary 4.1 The Concepts of Force and Mass A force is a push or a pull and is a vector quantity. Contact forces arise from the physical contact between two objects. Noncontact forces are also called action-at-a-distance forces, because they arise without physical contact between two objects. Mass is a property of matter that determines how difficult it is to acceler- ate or decelerate an object. Mass is a scalar quantity. 4.2 Newton’s First Law of Motion Newton’s first law of motion, some- times called the law of inertia, states that an object continues in a state of rest or in a state of motion at a constant velocity unless compelled to change that state by a net force. Inertia is the natural tendency of an object to remain at rest or in motion at a constant velocity. The mass of a body is a quantitative measure of inertia and is measured in an SI unit called the kilogram (kg). An inertial reference frame is one in which Newton’s law of inertia is valid. 4.3 Newton’s Second Law of Motion/4.4 The Vector Nature of Newton’s Second Law of Motion Newton’s second law of motion states that when a net force ΣF → acts on an object of mass m, the acceleration a → of the object can be obtained from Equation 4.1. This is a vector equation and, for motion in two dimensions, is equivalent to Equations 4.2a and 4.2b. In these equations the x and y subscripts refer to the scalar components of the force and acceler- ation vectors. The SI unit of force is the newton (N). Σ F → = ma → (4.1) Σ F x = ma x (4.2a) Σ F y = ma y (4.2b) When determining the net force, a free-body diagram is helpful. A free- body diagram is a diagram that represents the object and the forces acting on it.

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  Introduction to Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  This is a vector equation and, for motion in two dimensions, is equivalent to Equations 4.2a and 4.2b. In these equations the x and y subscripts refer to the scalar components of the force and acceleration vectors. The SI unit of force is the newton (N). When determining the net force, a free-body diagram is helpful. A free-body diagram is a diagram that represents the object and the forces acting on it. 4.5 Newton’s Third Law of Motion Newton’s third law of motion, often called the action–reaction law, states that whenever one object exerts a force on a second object, the second object exerts an oppositely directed force of equal magnitude on the first object. Check Your Understanding (The answers are given at the end of the book.) 23. A circus performer hangs stationary from a rope. She then begins to climb upward by pulling herself up, hand over hand. When she starts climbing, is the tension in the rope (a) less than, (b) equal to, or (c) greater than it is when she hangs stationary? 24. A freight train is accelerating on a level track. Other things being equal, would the tension in the coupling between the engine and the first car change if some of the cargo in the last car were transferred to any one of the other cars? 25. Two boxes have masses m 1 and m 2 , and m 2 is greater than m 1 . The boxes are being pushed across a frictionless horizontal surface. As the drawing shows, there are two possible arrangements, and the pushing force is the same in each. In which arrangement, (a) or (b), does the force that the left box applies to the right box have a greater magnitude, or (c) is the magnitude the same in both cases? m 1 m 1 m 2 (a) (b) m 2 Pushing force Pushing force undiminished along the rope. Then, a 540-N tension force T B acts upward on the left side of the scaffold pulley (see part a of the drawing). A tension force is also applied to the point P, where the rope attaches to the roof.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  100 Chapter 4 | Forces and Newton’s Laws of Motion happens when you walk. The physics of walking. Your foot exerts a force on the earth, and the earth exerts a reaction force on your foot. This reaction force is a static frictional force, and it opposes the impending backward motion of your foot, propelling you forward in the process. Kinetic friction can also cause an object to move, all the while opposing relative motion, as it does in Example 10. In this example the kinetic frictional force acts on the sled and opposes the relative motion of the sled and the earth. Newton’s third law indicates, however, that since the earth exerts the kinetic frictional force on the sled, the sled must exert a reaction force on the earth. In response, the earth accelerates, but because of the earth’s huge mass, the motion is too slight to be noticed. Check Your Understanding (The answers are given at the end of the book.) 14. Suppose that the coefficients of static and kinetic friction have values such that m s 5 1.4 m k for a crate in contact with a cement floor. Which one of the following statements is true? (a) The magnitude of the static frictional force is always 1.4 times the magnitude of the kinetic frictional force. (b) The magnitude of the kinetic frictional force is always 1.4 times the magnitude of the static frictional force. (c) The magnitude of the maximum static frictional force is 1.4 times the magnitude of the kinetic frictional force. 15. A person has a choice of either pushing or pulling a sled at a constant velocity, as the drawing illustrates. Friction is present. If the angle u is the same in both cases, does it require less force to push or to pull the sled? 16. A box has a weight of 150 N and is being pulled across a horizontal floor by a force that has a magnitude of 110 N. The pulling force can point horizontally, or it can point above the horizontal at an angle u.

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  Introduction to Thermal and Fluid Engineering

  • 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.

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