Chapter 17.3, Problem 3E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# A spring with a mass of 2 kg has damping constant 14, and a force of 6 N is required to keep the spring stretched 0.5 m beyond its natural length. The spring is stretched 1 m beyond its natural length and then released with zero velocity. Find the position of the mass at any time t.

To determine

To find: The position of the mass at any time t .

Explanation

Given data:

The spring is stretched beyond its natural length, so x=âˆ’0.5 , m=2â€‰kg , restoringâ€‰force=6â€‰N , dampingÂ constant=14

Formula used:

Write the expression for Hookeâ€™s Law.

restoringâ€‰force=âˆ’kx (1)

Here,

k is spring constant, and

x is difference between the natural length and length of due to force exerts.

Write the expression for damping force.

dampingâ€‰force=âˆ’cdxdt (2)

Here,

c is damping constant.

Write the expression for Newtonâ€™s Second Law.

md2xdt2+cdxdt+kx=0

mxâ€³+cxâ€²+kx=0 (3)

Write the expression for auxiliary equation.

mr2+cr+k=0 (4)

Write the expression for the Â roots.

r1=âˆ’câˆ’c2âˆ’4mk2m (5)

r2=âˆ’c+c2âˆ’4mk2m (6)

Write the expression for general solution of over damping case.

x(t)=c1er1t+c2er2t (7)

Substitute âˆ’0.5â€‰m for x and 6â€‰N for restoring force in equation (1),

6â€‰N=âˆ’k(âˆ’0.5â€‰m)k=6â€‰N0.5â€‰mk=12â€‰Nm

Substitute 12 for k , 14 for c and 2 for m in equation (3),

2xâ€³+14xâ€²+12x=0

Find the auxiliary equation using equation (4).

2r2+14r+12=0

Find the value of r1 using equation (5),

r1=âˆ’14âˆ’142âˆ’4(2)(12)2(2)=âˆ’14âˆ’196âˆ’964=âˆ’14âˆ’1004=âˆ’6

Find the value of r2 using equation (6),

r2=âˆ’14+142âˆ’4(2)(12)

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