Explanation Formula used: Undamped free vibrations: If the actual damping is very small or there is no damping, then corresponding differential equation is given by m u ″ + k u = 0 and the general solution is u = A cos ω 0 t + B sin ω 0 t where ω 0 2 = k m . For the equation u = A cos ω 0 t + B sin ω 0 t , the relationship between A, B and δ is given as follows. A = R cos δ , B = R sin δ where R = A 2 + B 2 tan δ = B A Calculation: Given that a mass weighing 3 lb stretches a spring 3 in. That is, ω = 3 l b and L = 3 in . Also it is given that the mass is pushed upward and then contracting the spring a distance of 1 in. From the above statement, it can be concluded that the mass is at –1 in distance from its actual point. Take this as initial condition of the motion of the mass. That is, u ( 0 ) = − 1 in . Since the motion with downward velocity is 2 ft/s , u ′ ( 0 ) = 2 ft/s . Since there is no damping, the motion of mass u should satisfy the differential equation m u ″ + k u = 0 with the initial conditions u ( 0 ) = − 1 12 ft and u ′ ( 0 ) = 2 ft/s . The solution is given by u = A cos ω 0 t + B sin ω 0 t where ω 0 2 = k m . In order to solve the differential equation m u ″ + k u = 0 , compute the values of m and k as follows. The value of m can be computed from the equation ω = m g as follows. ω = m g m = ω g m = 3 lb 32 ft / s 2 m = 3 32 lb s 2 f t − 1 The value of k can be computed from the equation ω − k L = 0 as follows. ω − k L = 0 k = ω L k = 3 lb 3 in k = 3 lb 1 4 ft k = 12 lb f t − 1 Substitute the obtained values of m and k in the differential equation m u ″ + k u = 0 and get the new equation 3 32 u ″ + 12 u = 0 . The value of ω 0 can be computed from the equation ω 0 2 = k m as follows. ω 0 2 = k m ω 0 2 = 12 3 32 ω 0 2 = 128 ω 0 = 8 2 Thus, the frequency of the motion is ω 0 = 8 2 r a d / s _ and the equation of motion is u = A cos 8 2 t + B sin 8 2 t . Apply the initial conditions u ( 0 ) = − 1 12 ft and u ′ ( 0 ) = 2 ft/s in the above equation and get the values of A and B as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 3.7, Problem 7P

To determine

The position u of the mass at time t. In addition, compute the frequency, period, phase and amplitude of the motion.

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Chapter 3.7, Problem 6P

Chapter 3.7, Problem 8P

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Chapter 3 Solutions

Elementary Differential Equations

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