Physics
Newton's Second Law in Angular Form
Newton's Second Law in Angular Form states that the torque acting on an object is equal to the product of the object's moment of inertia and its angular acceleration. Mathematically, it is expressed as τ = Iα, where τ represents torque, I is the moment of inertia, and α is the angular acceleration. This law describes the relationship between the forces and torques acting on rotating objects.
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10 Key excerpts on "Newton's Second Law in Angular Form"
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Thus, Equation 2.24 can be put in a simple but interesting form: arrowrightnosp arrowrightnosp N L t o o = d d . (2.25) That is, the time rate of change of angular momentum about a fixed point O is equal to the torque about the same point O. It further follows that if the torque of the force arrowrightnosp N O is zero, then the angular momentum arrowrightnosp L O is constant. The important concept of angular momentum will be found to be particularly useful in the study of central force motion as well as in the study of systems of particles and rigid bodies. We now apply the preceding result to a simple system: a particle of mass m moving on a circular path of radius r about the center O. In this case, we have v v r = = arrowrightnosp dotnosp θ and L L r p mr I = = × = = arrowrightnosp arrowrightnosp arrowrightnosp dotnosp dotnosp 2 θ θ where I = mr 2 is the moment of inertia of m about the axis of rotation. Newton’s second law for rotational motion now takes the simple form N I t I = = d d 2 2 θ α , which is a familiar result from basic physics. 2.6 WORK, ENERGY, AND CONSERVATION LAWS In the preceding sections, we looked at the three laws of Newtonian mechanics. In principle, once the forces are known, every detail of the motion can be predicted by the use of Newton’s second law of motion. Thus, the power of Newtonian mechanics lies in the possibility of finding force laws by which objects interact. For example, the force of air drag on an object can be obtained from a formula involving the size of the object and how fast it is moving; the force exerted by a spring depends on how Classical Mechanics 38 © 2010 Taylor & Francis Group, LLC much it is stretched or compressed. However, sometimes we are unable to discover in advance what forces will come into play when objects interact. Moreover, most problems in mechanics cannot be solved completely in terms of known force, or it is too tedious to solve them.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Why is there more angular acceleration in part (b)? In Section 8.5 we learned that the moment of inertia I of an object depends on both its mass and the distribution of mass (and the location of the axis of rotation). When the mass is concentrated far from the axis of rotation, the moment of inertia is larger than if it were concentrated closer Newton’s Second Law for Rotation | 249 to the axis. The moment of inertia I in part (a), therefore, is greater than that it is in part (b). All else being equal, then, the angular acceleration decreases as the moment of inertia increases. The following is the rotational equivalent to Newton’s second law (Equation 4.3.1): The moment of inertia I can be calculated using Equation 8.5.1 (for point masses) or Table 8.5.1 (for distributed masses). Equation 9.3.1 is sometimes called Newton’s second law for rotation because its role for rotational motion is analogous to ∑ = m F a for linear motion, where torque is analogous to force, moment of inertia is analogous to mass, and angular acceleration is analogous to linear acceleration. Newton’s Second Law for Rotation The net torque on a rigid body is related to its angular acceleration as follows: ∑ τ α = I (9.3.1) Going Deeper Derivation of Equation 9.3.1 for a Point Mass Consider a point mass m connected by a massless rod of length r to a point about which the rod is free to rotate. r m F The mass is acted on by a tangential force of magnitude F. Applying Newton’s second law to the tangential direction, we have = F ma t The tangential acceleration, however, is related to the angular acceleration by Equa- tion 8.4.2: α = a r t . Making this substitution into Newton’s second law we have α = F mr Now multiply both sides by r: α ( ) = rF mr 2 The left-hand side is the torque about the rotational axis—the net torque in this case because there is only one force. On the right-hand side, the factor mr 2 is the moment of inertia of the point mass (Equation 8.5.1).
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 - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
The net external force must be zero: a F S 5 0. 2. The net external torque must be zero: a t S 5 0. These two conditions, used in solving problems involving rotation in a plane—result in three equations and three unknowns—two from the first condition (corresponding to the x - and y -components of the force) and one from the sec- ond condition, on torques. These equations must be solved simultaneously. 8.4 The Rotational Second Law of Motion The moment of inertia of a group of particles is a of a group of particles is a I ; omr 2 [8.12] If a rigid object free to rotate about a fixed axis has a net external torque ot acting on it, then the object undergoes an angular acceleration a, where ot 5 I a a [8.13] This equation is the rotational equivalent of the second law of motion. Problems are solved by using Equation 8.13 together with Newton’s second law and solving the resulting equations simultaneously. The relation a 5 r a is often key in relating the translational equations to the rotational equations. REMARKS The angular momentum is unchanged by internal forces; however, the kinetic energy increases because the student must perform positive work to walk toward the center of the platform. QUESTION 8.18 Is energy conservation violated in this example? Explain why there is a positive net change in mechanical energy. What is the origin of this energy? EXERCISE 8.18 (a) Find the angular speed of the merry -go- round before the student jumped on, assuming the student didn’t transfer any momentum or energy as she jumped on the merry-go-round. (b) By how much did the kinetic energy of the system change when the student jumped on? Notice that energy is lost in this process, as should be expected, since it is essentially a per- r- r fectly inelastic collision.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 5 Dynamics Nature and nature's laws lay hid in night God said Let Newton be! And all was light. 5.1 The second law of Newton A. Pope According to everyday experience a net external force F ext must act on a material body in order to change its velocity by accelerating it. Thus force is the cause and acceleration is the result. Acceleration was defined a 2 in section 2.5. It is an empirical fact that for a given body its acceleration is proportional to the applied force, a <* F ext . Moreover, for a given force the more massive an object is, the smaller the resulting acceleration. These two experimental facts are summarized by the adjacent graph and are formulated by the second law of Newton, namely a = F ex t /w, where the mass of the body ra is defined by the ratio F ext /a. As indicated above, F ext is the resultant vector sum of all the external forces acting on the body, F ext = IF k , ex t so that the second law may be written in the usually stated form more massive F =YF = M ext ^ ' k,ext ma (5.1) *=i Mass is a scalar quantity, its unit in the SI system is the kilogram, kg. Since by (5.1) a given force produces a smaller acceleration in a larger mass, mass may be considered to quantify a body's resistance to a change in its velocity, in other words, its inertia. For this reason the mass which appears in the second law is termed inertial mass, to distinguish it from the role played by mass in gravitation (chapter 10). Mass is an invariant quantity, independent of the reference frame and has the same value on Earth, on any other planet, or in empty space. It is to be sharply distinguished from the weight of a body W, which is an expression of the gravitational force acting on it and therefore varies from place to place on Earth, and from planet to planet. The weight in Newtons is given by W = rag, where g is the local acceleration due gravity. So where g = 9.8 m/s 2 , a body of 129
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The center of gravity is closely related to the center-of-mass concept discussed in Section 7.5. To see why they are related, let’s replace each occurrence of the weight in Equation 9.3 by W 5 mg, where m is the mass of a given object and g is the magnitude of the acceleration due to gravity at the location of the object. Suppose that g has the same value everywhere the objects are located. Then it can be algebraically eliminated from each term on the right side of Equation 9.3. The resulting equation, which contains only masses and distances, is the same as Equation 7.10, which defines the location of the cen- ter of mass. Thus, the two points are identical. For ordinary-sized objects, like cars and boats, the center of gravity coincides with the center of mass. 9.4 | Newton’s Second Law for Rotational Motion About a Fixed Axis The goal of this section is to put Newton’s second law into a form suitable for describing the rotational motion of a rigid object about a fixed axis. We begin by considering a particle moving on a circular path. Figure 9.14 presents a good approximation of this situation by using a small model plane on a guideline of negligible mass. The plane’s engine produces a net external tangential force F T that gives the plane a tangential acceleration a T . In accord with Newton’s second law, it follows that F T 5 ma T . The torque t produced by this force is t 5 F T r, where the radius r of the circular path is also the lever arm. As a result, the torque is t 5 ma T r. However, the tangential acceleration is related to the angular acceleration a according to a T 5 ra (Equation 8.10), where a must be expressed in rad/s 2 . With this sub- stitution for a T , the torque becomes t 5 1 mr 2 2 a (9.4) u Moment of inertia I Equation 9.4 is the form of Newton’s second law we have been seeking. It indicates that the net external torque t is directly proportional to the angular acceleration a.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
We now need to take into account the possibility that a rigid object can also have an angular acceleration. A net external force causes linear motion to change, but what causes rotational motion to change? For example, something causes the rotational velocity of a speedboat’s propeller to change when the boat accelerates. Is it simply the net force? As it turns out, it is not the net external force, but rather the net external torque that causes the rotational velocity to change. Just as greater net forces cause greater linear accelerations, greater net torques cause greater rotational or angular accelerations. Figure 9.2 helps to explain the idea of torque. When you push on a door with a force F B , as in part a, the door opens more quickly when the force is larger. Other things being equal, a larger force generates a larger torque. However, the door does not open as quickly if you apply the same force at a point closer to the hinge, as in part b, because the force now produces less torque. Furthermore, if your push is directed nearly at the hinge, as in part c, you will have a hard time opening the door at all, because the torque is nearly zero. In summary, the torque depends on the magni- tude of the force, on the point where the force is applied relative to the axis of rotation (the hinge in Figure 9.2), and on the direction of the force. For simplicity, we deal with situations in which the force lies in a plane that is per- pendicular to the axis of rotation. In Figure 9.3, for instance, the axis is perpendicular to the page and the force lies in the plane of the paper. The drawing shows the line of action and the lever arm of the force, two concepts that are important in the definition of torque. The line of action is an extended line drawn colinear with the force. The lever arm is the distance < between the line of action and the axis of rotation, measured on a line that is 218 Mr.
- 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 - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
In pure translation there is no rotation of any line in the body. Because transla- tional motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9.1b. We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration. A net external force causes linear motion to change, but what causes rotational motion to change? For example, something causes the rotational velocity of a speedboat’s propeller to change when the boat accelerates. Is it simply the net force? As it turns out, it is not the net external force, but rather the net external torque that causes the rotational velocity to change. Just as greater net forces cause greater linear accelerations, greater net torques cause greater rotational or angular accelerations. Interactive Figure 9.2 helps to explain the idea of torque. When you push on a door with a force → F , as in part a, the door opens more quickly when the force is larger. Other things being equal, a larger force generates a larger torque. However, the door does not open as quickly if you apply the same force at a point closer to the hinge, as in part b, because the force now produces less torque. Furthermore, if your push is directed nearly at the hinge, as in part c, you will have a hard time opening the door at all, because the torque is nearly zero. In summary, the torque depends on the magnitude of the force, on the point where the force is applied relative to the axis of rotation (the hinge in Interac- tive Figure 9.2), and on the direction of the force. For simplicity, we deal with situations in which the force lies in a plane that is per- pendicular to the axis of rotation.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES After reading this module, you should be able to... 9.1 Distinguish between torque and force. 9.2 Analyze rigid objects in equilibrium. 9.3 Determine the center of gravity of rigid objects. 9.4 Analyze rotational dynamics using moments of inertia. 9.5 Apply the relation between rotational work and energy. 9.6 Solve problems using the conservation of angular momentum. Mr. Green/Shutterstock CHAPTER 9 Rotational Dynamics The large counterweight on the right side (short end) of this tall tower crane ensures its boom remains balanced on its mast while lifting heavy loads. It is not equal weights on both sides of the tower that keep it in equilibrium, but equal torques. Torque is the rotational analog of force, and is an important topic of this chapter. 9.1 The Action of Forces and Torques on Rigid Objects The mass of most rigid objects, such as a propeller or a wheel, is spread out and not concentrated at a single point. These objects can move in a number of ways. Figure 9.1a illustrates one possibility called translational motion, in which all points on the body travel on parallel paths (not necessarily straight lines). In pure translation there is no 223 Translation ( ) a Combined translation and rotation ( ) b FIGURE 9.1 Examples of (a) translational motion and (b) combined translational and rotational motions. 224 CHAPTER 9 Rotational Dynamics rotation of any line in the body. Because translational motion can occur along a curved line, it is often called curvilinear motion or linear motion. Another possibility is rotational motion, which may occur in combination with translational motion, as is the case for the somersaulting gymnast in Figure 9.1b. We have seen many examples of how a net force affects linear motion by causing an object to accelerate. We now need to take into account the possibility that a rigid object can also have an angular acceleration.
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