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

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Unformatted text preview: e your acceleration as you start sprinting from a standing position. Fictitious forces (a) (b) (c) 162 CHAPTER 6 EXAMPLE 6.9 Circular Motion and Other Applications of Newton’s Laws Fictitious Forces in Linear Motion A small sphere of mass m is hung by a cord from the ceiling of a boxcar that is accelerating to the right, as shown in Figure 6.13. According to the inertial observer at rest (Fig. 6.13a), the forces on the sphere are the force T exerted by the cord and the force of gravity. The inertial observer concludes that the acceleration of the sphere is the same as that of the boxcar and that this acceleration is provided by the horizontal component of T. Also, the vertical component of T balances the force of gravity. Therefore, she writes Newton’s second law as F T m g m a, which in component form becomes Inertial observer (1) Fx T sin Fy T cos Noninertial observer ma (2) Because the deﬂection of the cord from the vertical serves as a measure of acceleration, a simple pendulum can be used as an accelerometer. According to the noninertial observer riding in the car (Fig. 6.13b), the cord still makes an angle with the vertical; however, to her the sphere is at rest and so its acceleration is zero. Therefore, she introduces a ﬁctitious force to balance the horizontal component of T and claims that the net force on the sphere is zero! In this noninertial frame of reference, Newton’s second law in component form yields mg 0 Thus, by solving (1) and (2) simultaneously for a, the inertial observer can determine the magnitude of the car’s acceleration through the relationship a g tan Fx T sin Ffictitious Fy T cos mg Inertial observer Tθ mg (a) Noninertial observer Tθ mg (b) Figure 6.13 0 If we recognize that Fﬁctitious ma inertial ma, then these expressions are equivalent to (1) and (2); therefore, the noninertial observer obtains the same mathematical results as the inertial observer does. However, the physical interpretation of the deﬂection of the cord differs in the two frames of reference. a Fﬁctitious 0 A small sphere suspended from the ceiling of a boxcar accelerating to the right is deﬂected as shown. (a) An inertial observer at rest outside the car claims that the acceleration of the sphere is provided by the horizontal component of T. (b) A noninertial observer riding in the car says that the net force on the sphere is zero and that the deﬂection of the cord from the vertical is due to a ﬁctitious force Fﬁctitious that balances the horizontal component of T. 6.4 EXAMPLE 6.10 Motion in the Presence of Resistive Forces Fictitious Force in a Rotating System According to a noninertial observer attached to the turntable, the block is at rest and its acceleration is zero. Therefore, she must introduce a ﬁctitious outward force of magnitude mv 2/r to balance the inward force exerted by the string. According to her, the net force on the block is zero, and she writes Newton’s second law as T mv 2/r 0. Suppose a block of...
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## This document was uploaded on 09/19/2013.

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