Explanation Given: A block weighing 4 3 kg hangs to a spring that has spring constant k = 12 N/m . An external force F e x t = 9 cos ω t is produced due to the vibration of the spring. Results used: In a system with forced undamped oscillations if the force is a cosine curve then the equation is of the form y ′ ′ + ω 0 2 y = F 0 m cos ω t . The mass is m, F 0 is the amplitude of the external source, ω 0 is the natural frequency of the system and ω is the frequency of the external force. The general solution of the equation is y ( t ) = c 1 sin ω 0 t + c 2 cos ω 0 t + F 0 m ω 0 2 − ω 2 cos ω t . Calculation: Note that the given system is an example of forced undamped oscillations. In the given problem m = 4 3 , F 0 = 9 and ω 0 2 = k m . The value of ω 0 is calculated as follows. ω 0 = k m = 12 4 3 = 9 = 3 Therefore, ω 0 = 3 s − 1 . Use the values in the above result and obtain the general solution as follows. y ( t ) = c 1 sin ω 0 t + c 2 cos ω 0 t + F 0 m ω 0 2 − ω 2 cos ω t = c 1 sin 3 t + c 2 cos 3 t + 9 4 3 9 − ω 2 cos ω t = c 1 sin 3 t + c 2 cos 3 t + 6.75 9 − ω 2 cos ω t It is given that the initial displacement is 0 that is y ( 0 ) = 0 . Substitute t = 0 in above equation and obtain the following. y ( 0 ) = c 1 sin 3 ( 0 ) + c 2 cos 3 ( 0 ) + 6.75 9 − ω 2 cos ω 0 0 = c 1 ( 0 ) + c 2 ( 1 ) + 6.75 9 − ω 2 ( 1 ) c 2 = − 6.75 9 − ω 2 Hence, c 2 = − 6.75 9 − ω 2 . Differentiate y ( t ) with respect to t and obtain that y ′ ( t ) = 3 c 1 cos 3 t − 3 c 2 sin 3 t − 6.75 ω 9 − ω 2 sin ω t It is also given that the initial velocity is also 0, that is y ′ ( 0 ) = 0 . Substitute t = 0 in y ′ ( t ) = 3 c 1 cos 3 t − 3 c 2 sin 3 t − 6.75 ω 9 − ω 2 sin ω t and obtain the following. y ′ ( 0 ) = 3 c 1 cos 3 ( 0 ) − 3 c 2 sin 3 ( 0 ) − 6.75 ω 9 − ω 2 sin ω ( 0 ) 0 = 3 c 1 ( 1 ) − 3 c 2 ( 0 ) − 6.75 ω 9 − ω 2 ( 0 ) 0 = 3 c 1 c 1 = 0 Hence, c 1 = 0 . Using the values of c 1 and c 2 in general solution, the general solution can be written as follows. y ( t ) = ( 0 ) sin 3 t + ( − 6.75 9 − ω 2 ) cos 3 t + 6.75 9 − ω 2 cos ω t = 0 + 6.75 9 − ω 2 ( − cos 3 t + cos ω t ) = 6

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter D2.4, Problem 16E

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To graph: The solution of the initial value problem for the cases with ω=3.1 and ω=0.5 .

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Chapter D2.4, Problem 15E

Chapter D2.4, Problem 17E

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Chapter D2 Solutions

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Ch. D2.1 - Describe how to find the order of a differential...Ch. D2.1 - Prob. 2E Ch. D2.1 - Prob. 3E Ch. D2.1 - Give a general form of a second-order linear...Ch. D2.1 - Prob. 5E Ch. D2.1 - Prob. 6E Ch. D2.1 - Prob. 7E Ch. D2.1 - Prob. 8E Ch. D2.1 - Prob. 9E Ch. D2.1 - Prob. 10E

Ch. D2.1 - Prob. 11E Ch. D2.1 - Prob. 12E Ch. D2.1 - Prob. 13E Ch. D2.1 - Verifying solutions Verify by substitution that...Ch. D2.1 - Prob. 15E Ch. D2.1 - Prob. 16E Ch. D2.1 - Prob. 17E Ch. D2.1 - Prob. 18E Ch. D2.1 - Prob. 19E Ch. D2.1 - Prob. 20E Ch. D2.1 - Prob. 21E Ch. D2.1 - Prob. 22E Ch. D2.1 - Prob. 23E Ch. D2.1 - Prob. 24E Ch. D2.1 - Prob. 25E Ch. D2.1 - Prob. 26E Ch. D2.1 - Prob. 27E Ch. D2.1 - Prob. 28E Ch. D2.1 - Prob. 29E Ch. D2.1 - Prob. 30E Ch. D2.1 - Prob. 31E Ch. D2.1 - Prob. 32E Ch. D2.1 - Prob. 33E Ch. D2.1 - Prob. 34E Ch. D2.1 - Prob. 35E Ch. D2.1 - Prob. 36E Ch. D2.1 - Prob. 37E Ch. D2.1 - Prob. 38E Ch. D2.1 - Prob. 39E Ch. D2.1 - Prob. 40E Ch. D2.1 - Prob. 41E Ch. D2.1 - Prob. 42E Ch. D2.1 - Prob. 43E Ch. D2.1 - Initial value problems Solve the following initial...Ch. D2.1 - Prob. 45E Ch. D2.1 - Prob. 46E Ch. D2.1 - Explain why or why not Determine whether the...Ch. D2.1 - Prob. 48E Ch. D2.1 - Solution verification Verify by substitution that...Ch. D2.1 - Prob. 50E Ch. D2.1 - Prob. 51E Ch. D2.1 - Prob. 52E Ch. D2.1 - Prob. 53E Ch. D2.1 - Prob. 54E Ch. D2.1 - Prob. 55E Ch. D2.1 - Prob. 56E Ch. D2.1 - Prob. 57E Ch. D2.1 - Prob. 58E Ch. D2.1 - Prob. 59E Ch. D2.1 - Prob. 60E Ch. D2.1 - Prob. 61E Ch. D2.1 - Prob. 62E Ch. D2.1 - Prob. 63E Ch. D2.1 - Prob. 64E Ch. D2.1 - Prob. 65E Ch. D2.1 - Prob. 66E Ch. D2.1 - Prob. 67E Ch. D2.1 - Prob. 68E Ch. D2.1 - Prob. 69E Ch. D2.1 - Reduction of order Suppose you are solving a...Ch. D2.2 - Prob. 1E Ch. D2.2 - Prob. 2E Ch. D2.2 - Prob. 3E Ch. D2.2 - Prob. 4E Ch. D2.2 - Prob. 5E Ch. D2.2 - Prob. 6E Ch. D2.2 - Prob. 7E Ch. D2.2 - Give the trial solution used to solve a...Ch. D2.2 - Prob. 9E Ch. D2.2 - Prob. 10E Ch. D2.2 - General solutions with distinct real roots Find...Ch. D2.2 - Prob. 12E Ch. D2.2 - Prob. 13E Ch. D2.2 - Prob. 14E Ch. D2.2 - Initial value problems with distinct real roots...Ch. D2.2 - Prob. 16E Ch. D2.2 - Prob. 17E Ch. D2.2 - Prob. 18E Ch. D2.2 - Prob. 19E Ch. D2.2 - Prob. 20E Ch. D2.2 - Prob. 21E Ch. D2.2 - Prob. 22E Ch. D2.2 - Prob. 23E Ch. D2.2 - Prob. 24E Ch. D2.2 - Prob. 25E Ch. D2.2 - Prob. 26E Ch. D2.2 - Prob. 27E Ch. D2.2 - Prob. 28E Ch. D2.2 - Prob. 29E Ch. D2.2 - Prob. 30E Ch. D2.2 - Prob. 31E Ch. D2.2 - Prob. 32E Ch. D2.2 - Prob. 33E Ch. D2.2 - Prob. 34E Ch. D2.2 - Initial value problems with Cauchy-Euler equations...Ch. D2.2 - Prob. 36E Ch. D2.2 - Prob. 37E Ch. D2.2 - Initial value problems with Cauchy-Euler equations...Ch. D2.2 - Prob. 39E Ch. D2.2 - Prob. 42E Ch. D2.2 - Prob. 43E Ch. D2.2 - Prob. 44E Ch. D2.2 - Prob. 45E Ch. D2.2 - Prob. 46E Ch. D2.2 - Prob. 47E Ch. D2.2 - Prob. 48E Ch. D2.2 - Prob. 49E Ch. D2.2 - Prob. 50E Ch. D2.2 - Prob. 51E Ch. D2.2 - Cauchy-Euler equation with repeated roots It can...Ch. D2.2 - Prob. 53E Ch. D2.2 - Prob. 54E Ch. D2.2 - Prob. 55E Ch. D2.2 - Prob. 56E Ch. D2.2 - Prob. 57E Ch. D2.2 - Prob. 58E Ch. D2.2 - Prob. 59E Ch. D2.2 - Prob. 60E Ch. D2.2 - Prob. 61E Ch. D2.2 - Cauchy-Euler equation with repeated roots One of...Ch. D2.2 - Prob. 63E Ch. D2.2 - Prob. 64E Ch. D2.2 - Prob. 65E Ch. D2.2 - Prob. 66E Ch. D2.3 - Explain how to find the general solution of the...Ch. D2.3 - Prob. 2E Ch. D2.3 - Prob. 3E Ch. D2.3 - Prob. 4E Ch. D2.3 - Prob. 5E Ch. D2.3 - Prob. 6E Ch. D2.3 - Prob. 7E Ch. D2.3 - Prob. 8E Ch. D2.3 - Prob. 9E Ch. D2.3 - Prob. 10E Ch. D2.3 - Prob. 11E Ch. D2.3 - Prob. 12E Ch. D2.3 - Prob. 13E Ch. D2.3 - Undetermined coefficients with exponentials Find a...Ch. D2.3 - Prob. 15E Ch. D2.3 - Prob. 16E Ch. D2.3 - Prob. 17E Ch. D2.3 - Prob. 18E Ch. D2.3 - Prob. 19E Ch. D2.3 - Prob. 20E Ch. D2.3 - Prob. 21E Ch. D2.3 - Prob. 22E Ch. D2.3 - Prob. 23E Ch. D2.3 - Prob. 24E Ch. D2.3 - Prob. 25E Ch. D2.3 - Prob. 26E Ch. D2.3 - Prob. 27E Ch. D2.3 - Prob. 28E Ch. D2.3 - Prob. 29E Ch. D2.3 - Prob. 30E Ch. D2.3 - Prob. 31E Ch. D2.3 - Prob. 32E Ch. D2.3 - Prob. 33E Ch. D2.3 - Prob. 34E Ch. D2.3 - Prob. 35E Ch. D2.3 - Prob. 36E Ch. D2.3 - Prob. 37E Ch. D2.3 - Initial value problems Find the general solution...Ch. D2.3 - Prob. 39E Ch. D2.3 - Prob. 40E Ch. D2.3 - Prob. 41E Ch. D2.3 - Prob. 42E Ch. D2.3 - Prob. 43E Ch. D2.3 - Prob. 44E Ch. D2.3 - Prob. 45E Ch. D2.3 - Prob. 46E Ch. D2.3 - Prob. 47E Ch. D2.3 - Prob. 48E Ch. D2.3 - Prob. 49E Ch. D2.3 - Prob. 50E Ch. D2.3 - Prob. 51E Ch. D2.3 - Variation of parameters Finding a particular...Ch. D2.4 - Explain the meaning of the words damped, undamped,...Ch. D2.4 - In the models discussed in this section, under...Ch. D2.4 - Prob. 3E Ch. D2.4 - Prob. 4E Ch. D2.4 - Prob. 5E Ch. D2.4 - Prob. 6E Ch. D2.4 - Prob. 7E Ch. D2.4 - Prob. 8E Ch. D2.4 - Prob. 9E Ch. D2.4 - Free undamped oscillations Solve the initial value...Ch. D2.4 - Prob. 11E Ch. D2.4 - Prob. 12E Ch. D2.4 - Prob. 13E Ch. D2.4 - Prob. 14E Ch. D2.4 - Prob. 15E Ch. D2.4 - Prob. 16E Ch. D2.4 - Free damped oscillations Solve the initial value...Ch. D2.4 - Free damped oscillations Solve the initial value...Ch. D2.4 - Designing a shock absorber A shock absorber must...Ch. D2.4 - Designing a suspension system A spring in a...Ch. D2.4 - Forced damped oscillations 21.A 1-kg block hangs...Ch. D2.4 - Forced damped oscillations 22.A 20-kg block hangs...Ch. D2.4 - Prob. 23E Ch. D2.4 - Prob. 24E Ch. D2.4 - Prob. 25E Ch. D2.4 - Prob. 26E Ch. D2.4 - Prob. 27E Ch. D2.4 - LCR circuits 28.The circuit in Exercise 27 (10-ohm...Ch. D2.4 - Prob. 29E Ch. D2.4 - Prob. 30E Ch. D2.4 - Prob. 31E Ch. D2.4 - LCR circuits 32.Find the charge on the capacitor...Ch. D2.4 - Explain why or why not Determine whether the...Ch. D2.4 - Prob. 34E Ch. D2.4 - Prob. 35E Ch. D2.4 - Prob. 36E Ch. D2.4 - Prob. 37E Ch. D2.4 - Prob. 38E Ch. D2.4 - Prob. 39E Ch. D2.4 - Prob. 41E Ch. D2.4 - Prob. 42E Ch. D2.4 - Prob. 43E Ch. D2.4 - Prob. 44E Ch. D2.4 - Applications 4346.Horizontal oscillators The...Ch. D2.4 - Prob. 46E Ch. D2.4 - Prob. 47E Ch. D2.4 - Prob. 48E Ch. D2.4 - Prob. 49E Ch. D2.4 - Prob. 51E Ch. D2.4 - Prob. 52E Ch. D2.5 - Prob. 1E Ch. D2.5 - Prob. 2E Ch. D2.5 - Prob. 3E Ch. D2.5 - Prob. 4E Ch. D2.5 - Prob. 5E Ch. D2.5 - Prob. 6E Ch. D2.5 - Prob. 7E Ch. D2.5 - Prob. 8E Ch. D2.5 - Gain and phase lag functions Consider the...Ch. D2.5 - Prob. 10E Ch. D2.5 - Prob. 11E Ch. D2.5 - Solutions to oscillator equations Consider the...Ch. D2.5 - Prob. 13E Ch. D2.5 - Solutions to oscillator equations Consider the...Ch. D2.5 - Prob. 15E Ch. D2.5 - Prob. 16E Ch. D2.5 - Prob. 17E Ch. D2.5 - Prob. 18E Ch. D2.5 - Analyzing circuit equations Consider the circuit...Ch. D2.5 - Prob. 20E Ch. D2.5 - Prob. 21E Ch. D2.5 - Prob. 22E Ch. D2.5 - Prob. 23E Ch. D2.5 - A high-pass filter Consider the LCR circuit shown...Ch. D2.5 - High-pass filters Consider the high-pass filter...Ch. D2.5 - Prob. 26E Ch. D2.5 - High-pass filters Consider the high-pass filter...Ch. D2.5 - Prob. 28E Ch. D2 - Prob. 1RE Ch. D2 - Prob. 2RE Ch. D2 - Prob. 3RE Ch. D2 - Prob. 4RE Ch. D2 - Solving homogeneous equations Find the general...Ch. D2 - Prob. 6RE Ch. D2 - Prob. 7RE Ch. D2 - Prob. 8RE Ch. D2 - Prob. 9RE Ch. D2 - Prob. 10RE Ch. D2 - Prob. 11RE Ch. D2 - Prob. 12RE Ch. D2 - Prob. 13RE Ch. D2 - Prob. 14RE Ch. D2 - Prob. 15RE Ch. D2 - Prob. 16RE Ch. D2 - Prob. 17RE Ch. D2 - Prob. 18RE Ch. D2 - Prob. 19RE Ch. D2 - Prob. 20RE Ch. D2 - Prob. 21RE Ch. D2 - Forced undamped oscillations A 4-kg block hangs on...Ch. D2 - Free damped oscillations A 0.2-kg block hangs on a...

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To graph: The solution of the initial value problem for the cases with ω = 3.1 and ω = 0.5 .

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To graph: The solution of the initial value problem for the cases with ω=3.1 and ω=0.5 .

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Chapter D2 Solutions