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6 - Circular Motion and Other Applications of Newton's Laws

In a rotating system according to a noninertial

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Unformatted text preview: mass m lying on a horizontal, frictionless turntable is connected to a string attached to the center of the turntable, as shown in Figure 6.14. According to an inertial observer, if the block rotates uniformly, it undergoes an acceleration of magnitude v 2/r, where v is its linear speed. The inertial observer concludes that this centripetal acceleration is provided by the force T exerted by the string and writes Newton’s second law as T mv 2/r. n Noninertial observer n T Ffictitious = mg T mv 2 r mg (a) Inertial observer (b) Figure 6.14 A block of mass m connected to a string tied to the center of a rotating turntable. (a) The inertial observer claims that the force causing the circular motion is provided by the force T exerted by the string on the block. (b) The noninertial observer claims that the block is not accelerating, and therefore she introduces a fictitious force of magnitude mv 2/r that acts outward and balances the force T. Optional Section 6.4 4.9 163 MOTION IN THE PRESENCE OF RESISTIVE FORCES In the preceding chapter we described the force of kinetic friction exerted on an object moving on some surface. We completely ignored any interaction between the object and the medium through which it moves. Now let us consider the effect of that medium, which can be either a liquid or a gas. The medium exerts a resistive force R on the object moving through it. Some examples are the air resistance associated with moving vehicles (sometimes called air drag ) and the viscous forces that act on objects moving through a liquid. The magnitude of R depends on such factors as the speed of the object, and the direction of R is always opposite the direction of motion of the object relative to the medium. The magnitude of R nearly always increases with increasing speed. The magnitude of the resistive force can depend on speed in a complex way, and here we consider only two situations. In the first situation, we assume the resistive force is proportional to the speed of the moving object; this assumption is valid for objects falling slowly through a liquid and for very small objects, such as dust particles, moving through air. In the second situation, we assume a resistive force that is proportional to the square of the speed of the moving object; large objects, such as a skydiver moving through air in free fall, experience such a force. 164 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws v=0 a=g v R vt v 0.63vt mg (a) t τ v = vt a=0 (c) (b) Figure 6.15 (a) A small sphere falling through a liquid. (b) Motion diagram of the sphere as it falls. (c) Speed – time graph for the sphere. The sphere reaches a maximum, or terminal, speed vt , and the time constant is the time it takes to reach 0.63vt . Resistive Force Proportional to Object Speed If we assume that the resistive force acting on an object moving through a liquid or gas is proportional to the object’s speed, then the magnitude of the resistive force can be expressed as...
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