Circular Motion and Other Applications of Newton's Laws

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168 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws such a way that the front-facing surface area does not increase. Determine the relationship between the resistive force exerted by the air and the speed of the falling filters. Solution At terminal speed, the upward resistive force balances the downward force of gravity. So, a single filter falling at its terminal speed experiences a resistive force of 1.64 g R � mg �� 1000 g/kg� (9.80 m/s2 ) � 0.016 1 N Resistive force (N) TABLE 6.2 Terminal Speed for Stacked Coffee Filters Number v t of Filters (m/s) a 1 1.01 2 1.40 3 1.63 4 2.00 5 2.25 6 2.40 7 2.57 8 2.80 9 3.05 10 3.22 a All values of vt are approximate. 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 1 2 3 4 Terminal speed (m/s) (a) Two filters nested together experience 0.032 2 N of resistive force, and so forth. A graph of the resistive force on the filters as a function of terminal speed is shown in Figure 6.17a. A straight line would not be a good fit, indicating that the resistive force is not proportional to the speed. The curved line is for a second-order polynomial, indicating a proportionality of the resistive force to the square of the speed. This proportionality is more clearly seen in Figure 6.17b, in which the resistive force is plotted as a function of the square of the terminal speed. 0 2 6 12 Terminal speed squared (m/s) 2 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 4 8 10 Figure 6.17 (a) Relationship between the resistive force acting on falling coffee filters and their terminal speed. The curved line is a second-order polynomial fit. (b) Graph relating the resistive force to the square of the terminal speed. The fit of the straight line to the data points indicates that the resistive force is proportional to the terminal speed squared. Can you find the proportionality constant? Resistive force (N) Pleated coffee filters can be nested together so that the force of air resistance can be studied. ( (b)
Optional Section 6.5 EXAMPLE 6.14 NUMERICAL MODELING IN PARTICLE DYNAMICS 2 6.5 Numerical Modeling in Particle Dynamics 169 Resistive Force Exerted on a Baseball A pitcher hurls a 0.145-kg baseball past a batter at 40.2 m/s (�90 mi/h). Find the resistive force acting on the ball at this speed. Solution We do not expect the air to exert a huge force on the ball, and so the resistive force we calculate from Equation 6.6 should not be more than a few newtons. First, we must determine the drag coefficient D. We do this by imagining that we drop the baseball and allow it to reach terminal speed. We solve Equation 6.9 for D and substitute the appropriate values for m, v t, and A from Table 6.1. Taking the density of air as 1.29 kg/m 3 , we obtain � 0.284 As we have seen in this and the preceding chapter, the study of the dynamics of a particle focuses on describing the position, velocity, and acceleration as functions of time. Cause-and-effect relationships exist among these quantities: Velocity causes position to change, and acceleration causes velocity to change. Because acceleration is the direct result of applied forces, any analysis of the dynamics of a particle usually begins with an evaluation of the net force being exerted on the particle. Up till now, we have used what is called the analytical method to investigate the position, velocity, and acceleration of a moving particle. Let us review this method briefly before learning about a second way of approaching problems in dynamics. (Because we confine our discussion to one-dimensional motion in this section, boldface notation will not be used for vector quantities.) If a particle of mass m moves under the influence of a net force �F, Newton’s second law tells us that the acceleration of the particle is a ��F/m. In general, we apply the analytical method to a dynamics problem using the following procedure: 1. Sum all the forces acting on the particle to get the net force �F. 2. Use this net force to determine the acceleration from the relationship a ��F/m. 3. Use this acceleration to determine the velocity from the relationship dv/dt � a. 4. Use this velocity to determine the position from the relationship dx/dt � v. The following straightforward example illustrates this method. EXAMPLE 6.15 D � This number has no dimensions. We have kept an extra digit beyond the two that are significant and will drop it at the end of our calculation. We can now use this value for D in Equation 6.6 to find the magnitude of the resistive force: � 1 2 (0.284)(1.29 kg/m3 )(4.2 � 10�3 m2 )(40.2 m/s) 2 R � 1 2 D�Av2 � 2 mg � v 2 t �A 1.2 N An Object Falling in a Vacuum — Analytical Method Consider a particle falling in a vacuum under the influence of the force of gravity, as shown in Figure 6.18. Use the analytical method to find the acceleration, velocity, and position of the particle. 2 The authors are most grateful to Colonel James Head of the U.S. Air Force Academy for preparing this section. See the Student Tools CD-ROM for some assistance with numerical modeling. 2(0.145 kg)(9.80 m/s 2 ) (43 m/s) 2 (1.29 kg/m 3 )(4.2 � 10 �3 m 2 ) Solution The only force acting on the particle is the downward force of gravity of magnitude F g, which is also the net force. Applying Newton’s second law, we set the net force acting on the particle equal to the mass of the particle times
Page 1 and 2: Circular Motion and Other Applicati
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Circular Motion and Other Applications of Newton's Laws

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