49. Show that Equation 15.32 is a solution of Equation 15.31 provided that b2< 4mk. One common type of retarding force is that discussed in Section 6.4, where the force is proportional to the speed of the moving object and acts in the direc- tion opposite the velocity of the object with respect to the medium. This retarding force is often observed when an object moves through air, for instance. Because the retarding force can be expressed as R = -bv (where bis a constant called the damping coefficient) and the restoring force of the system is -kx, we can write New- ton's second law as SF,= -kx - bv, = ma, dx d'x -kx - b = m (15.31) dt di" The solution to this equation requires mathematics that may be unfamiliar to you; we simply state it here without proof. When the retarding force is small compared with the maximum restoring force-that is, when the damping coefficient b is small-the solution to Equation 15.31 is x = Ae 2w cos (ot + o) (15.32)

49. Show that Equation 15.32 is a solution of Equation 15.31 provided that b2< 4mk. One common type of retarding force is that discussed in Section 6.4, where the force is proportional to the speed of the moving object and acts in the direc- tion opposite the velocity of the object with respect to the medium. This retarding force is often observed when an object moves through air, for instance. Because the retarding force can be expressed as R = -bv (where bis a constant called the damping coefficient) and the restoring force of the system is -kx, we can write New- ton's second law as SF,= -kx - bv, = ma, dx d'x -kx - b = m (15.31) dt di" The solution to this equation requires mathematics that may be unfamiliar to you; we simply state it here without proof. When the retarding force is small compared with the maximum restoring force-that is, when the damping coefficient b is small-the solution to Equation 15.31 is x = Ae 2w cos (ot + o) (15.32)

Physics for Scientists and Engineers, Technology Update (No access codes included)

9th Edition

ISBN:9781305116399

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter14: Fluid Mechanics

Section: Chapter Questions

Problem 14.30P: The United States possesses the ten largest warships in the world, aircraft carriers of the Nimitz...

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