(a) Explanation Given: The mass of the block is m spring constant is k the damping constant is c and F e x t is the external force. The position of the block is denoted by x ( t ) where x increases to the right and x = 0 is the position at which string is neither stretched nor compressed. Results used: The general differential equation for a motion of a string is m x ′ ′ + c x ′ + k x = F e x t . Calculation: For the given condition the differential equation is 2 x ′ ′ + 6 x ′ + 8 x = 0 . It can be expressed as x ′ ′ + 3 x ′ + 4 x = 0 . The characteristic polynomial is r 2 + 3 r + 4 = 0 . The roots are given as follows. r = − 3 ± 9 − 4 ( 1 ) ( 4 ) 2 = − 3 ± − 7 2 = − 3 2 ± 7 2 i Comparing with a ± b i obtain that a = − 3 2 , b = 7 2 . Thus the general solution is given by x ( t ) = e − 3 t 2 ( c 1 cos 7 2 t + c 2 sin 7 2 t ) . The initial condition given is x ( 0 ) = 2 . Substitute t = 0 in x ( t ) = e − 3 t 2 ( c 1 cos 7 2 t + c 2 sin 7 2 t ) and obtain the following. x ( 0 ) = e − 3 ( 0 ) 2 ( c 1 cos 7 2 ( 0 ) + c 2 sin 7 2 ( 0 ) ) 2 = c 1 Therefore, c 1 = 2 (b) To determine To graph: The position functions on the coordinate axes. (c) To determine To describe: The type of motion observed in the graph.

Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

2nd Edition

ISBN: 9780321954329

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter D2.4, Problem 43E

(a)

To determine

To find: The position of the block for m=2 kg,c=6 kg/s,k=8 N/m,Fext=0,x(0)=2,x′(0)=−1 .

(b)

To determine

To graph: The position functions on the coordinate axes.

(c)

To determine

To describe: The type of motion observed in the graph.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter D2.4, Problem 42E

Chapter D2.4, Problem 44E

Chapter D2 Solutions

Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

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...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The position of the block for m = 2 kg, c = 6 kg/s, k = 8 N/m, F e x t = 0 , x ( 0 ) = 2 , x ′ ( 0 ) = − 1 .

Solution Summary: The author explains the differential equation for the position of the block for x(t).

Videos

To find: The position of the block for m=2 kg,c=6 kg/s,k=8 N/m,Fext=0,x(0)=2,x′(0)=−1 .

To graph: The position functions on the coordinate axes.

To describe: The type of motion observed in the graph.

Want to see the full answer?

Chapter D2 Solutions