Chapter 17.3, Problem 4E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# A force of 13 N is needed to keep a spring with a 2-kg mass stretched 0.25 m beyond its natural length. The damping constant of the spring is c = 8.(a) If the mass starts at the equilibrium position with a velocity of 0.5 m/s, find its position at time t.(b) Graph the position function of the mass.

(a)

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’0.25 , m=2â€‰kg , restoringâ€‰force=13â€‰N , dampingÂ constant=8

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 roots.

r1r2}=âˆ’c2mÂ±Ï‰i (5)

Here,

Ï‰=4mkâˆ’c22m (6)

Write the expression for general solution of under damping case.

x(t)=eâˆ’(c2m)t(c1cosÏ‰t+c2sinÏ‰t) (7)

Substitute âˆ’0.25â€‰m for x and 13â€‰N for restoring force in equation (1),

13â€‰N=âˆ’k(âˆ’0.25â€‰m)k=13â€‰N0.25â€‰mk=52â€‰Nm

Substitute 52 for k , 8 for c and 2 for m in equation (3),

2xâ€³+8xâ€²+52x=0

Find the auxiliary equation using equation (4).

2r2+8r+52=0

Find the value of Ï‰ using equation (6).

Substitute 52 for k , 8 for c and 2 for m in equation (6),

Ï‰=4(2)(52)âˆ’(8)22(2)=3524=16Ã—224=4224

Simplify Ï‰ as follows.

Ï‰=22

Find the value of c2m .

c2m=82(2)=2

Substitute 22 for Ï‰ , and 2 for c2m in equation (7),

x(t)=eâˆ’(2)t(c1cos22t+c2sin22t) (8)

Since, the spring starts at equilibrium the value of x(0)=0

(b)

To determine

To graph: The position function of the mass.

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