Explanation The general expression for Damped oscillator. − k x − b d x d t = m d 2 x d t 2 (I) Here, k is the force constant, b is the damping constant, m is the mass. Let the solution of the equation (I) as, x = A e − b t / 2 m cos ( ω t + ϕ ) (II) Here, A is the amplitude, ϕ is the phase angle. Differentiate equation (II) with respect to time, d x d t = A e − b t / 2 m ( − b 2 m ) cos ( ω t + ϕ ) − A e − b t / 2 m ω sin ( ω t + ϕ ) (III) Differentiate equation (III) with respect to time d 2 x d t 2 = − b 2 m [ A e − b t / 2 m ( − b 2 m ) cos ( ω t + ϕ ) − A e − b t / 2 m ω sin ( ω t + ϕ ) ] − [ A e − b t / 2 m ( − b 2 m ) ω sin ( ω t + ϕ ) + A e − b t / 2 m ω 2 cos ( ω t + ϕ ) ] (IV) Use equation (II), (III) and (IV) in equation (I), − k A e − b t / 2 m cos ( ω t + ϕ ) + b 2 2 m A e − b t / 2 m cos ( ω<

Principles of Physics: A Calculus-Based Text

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Author: Raymond A. Serway, John W. Jewett

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Chapter 12, Problem 38P

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Show that equation x=Ae−bt/2mcos(ωt+ϕ) is a solution of −kx−bdxdt=md2xdt2.

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Chapter 12, Problem 37P

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Chapter 12 Solutions

Principles of Physics: A Calculus-Based Text

Ch. 12.1 - A block on the end of a spring is pulled to...Ch. 12.2 - Consider a graphical representation (Fig. 12.3) of...Ch. 12.2 - Figure 12.4 shows two curves representing...Ch. 12.2 - An object of mass m is hung from a spring and set...Ch. 12.4 - A grandfather clock depends on the period of a...Ch. 12.5 - Prob. 12.6QQ Ch. 12 - Which of the following statements is not true...Ch. 12 - Prob. 2OQ Ch. 12 - Prob. 3OQ Ch. 12 - Prob. 4OQ

Ch. 12 - Prob. 5OQ Ch. 12 - Prob. 6OQ Ch. 12 - If a simple pendulum oscillates with small...Ch. 12 - Prob. 8OQ Ch. 12 - Prob. 9OQ Ch. 12 - Prob. 10OQ Ch. 12 - Prob. 11OQ Ch. 12 - Prob. 12OQ Ch. 12 - Prob. 13OQ Ch. 12 - You attach a block to the bottom end of a spring...Ch. 12 - Prob. 15OQ Ch. 12 - Prob. 1CQ Ch. 12 - The equations listed in Table 2.2 give position as...Ch. 12 - Prob. 3CQ Ch. 12 - Prob. 4CQ Ch. 12 - Prob. 5CQ Ch. 12 - Prob. 6CQ Ch. 12 - The mechanical energy of an undamped blockspring...Ch. 12 - Prob. 8CQ Ch. 12 - Prob. 9CQ Ch. 12 - Prob. 10CQ Ch. 12 - Prob. 11CQ Ch. 12 - Prob. 12CQ Ch. 12 - Consider the simplified single-piston engine in...Ch. 12 - A 0.60-kg block attached to a spring with force...Ch. 12 - When a 4.25-kg object is placed on top of a...Ch. 12 - The position of a particle is given by the...Ch. 12 - You attach an object to the bottom end of a...Ch. 12 - A 7.00-kg object is hung from the bottom end of a...Ch. 12 - Prob. 6P Ch. 12 - Prob. 7P Ch. 12 - Prob. 8P Ch. 12 - Prob. 9P Ch. 12 - A 1.00-kg glider attached to a spring with a force...Ch. 12 - Prob. 11P Ch. 12 - Prob. 12P Ch. 12 - A 500-kg object attached to a spring with a force...Ch. 12 - In an engine, a piston oscillates with simple...Ch. 12 - A vibration sensor, used in testing a washing...Ch. 12 - A blockspring system oscillates with an amplitude...Ch. 12 - A block of unknown mass is attached to a spring...Ch. 12 - Prob. 18P Ch. 12 - Prob. 19P Ch. 12 - A 200-g block is attached to a horizontal spring...Ch. 12 - A 50.0-g object connected to a spring with a force...Ch. 12 - Prob. 22P Ch. 12 - Prob. 23P Ch. 12 - Prob. 24P Ch. 12 - Prob. 25P Ch. 12 - Prob. 26P Ch. 12 - Prob. 27P Ch. 12 - Prob. 28P Ch. 12 - The angular position of a pendulum is represented...Ch. 12 - A small object is attached to the end of a string...Ch. 12 - A very light rigid rod of length 0.500 m extends...Ch. 12 - A particle of mass m slides without friction...Ch. 12 - Review. A simple pendulum is 5.00 m long. What is...Ch. 12 - Prob. 34P Ch. 12 - Prob. 35P Ch. 12 - Show that the time rate of change of mechanical...Ch. 12 - Prob. 37P Ch. 12 - Prob. 38P Ch. 12 - Prob. 39P Ch. 12 - Prob. 40P Ch. 12 - Prob. 41P Ch. 12 - Prob. 42P Ch. 12 - Prob. 43P Ch. 12 - Prob. 44P Ch. 12 - Four people, each with a mass of 72.4 kg, are in a...Ch. 12 - Prob. 46P Ch. 12 - Prob. 47P Ch. 12 - Prob. 48P Ch. 12 - Prob. 49P Ch. 12 - Prob. 50P Ch. 12 - Prob. 51P Ch. 12 - Prob. 52P Ch. 12 - Prob. 53P Ch. 12 - Prob. 54P Ch. 12 - Prob. 55P Ch. 12 - A block of mass m is connected to two springs of...Ch. 12 - Review. One end of a light spring with force...Ch. 12 - Prob. 58P Ch. 12 - A small ball of mass M is attached to the end of a...Ch. 12 - Prob. 60P Ch. 12 - Prob. 61P Ch. 12 - Prob. 62P Ch. 12 - Prob. 63P Ch. 12 - A smaller disk of radius r and mass m is attached...Ch. 12 - A pendulum of length L and mass M has a spring of...Ch. 12 - Consider the damped oscillator illustrated in...Ch. 12 - An object of mass m1 = 9.00 kg is in equilibrium...Ch. 12 - Prob. 68P Ch. 12 - A block of mass M is connected to a spring of mass...

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Show that equation x = A e − b t / 2 m cos ( ω t + ϕ ) is a solution of − k x − b d x d t = m d 2 x d t 2 .

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