(a) Explanation Given data: The spring is stretched beyond its natural length, so x = − 0.25 , m = 2 kg , restoring force = 13 N , damping constant = 8 Formula used: Write the expression for Hooke’s Law. restoring force = − k x (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. damping force = − c d x d t (2) Here, c is damping constant. Write the expression for Newton’s Second Law. m d 2 x d t 2 + c d x d t + k x = 0 m x ″ + c x ′ + k x = 0 (3) Write the expression for auxiliary equation. m r 2 + c r + k = 0 (4) Write the expression for the roots. r 1 r 2 } = − c 2 m ± ω i (5) Here, ω = 4 m k − c 2 2 m (6) Write the expression for general solution of under damping case. x ( t ) = e − ( c 2 m ) t ( c 1 cos ω t + c 2 sin ω t ) (7) Substitute − 0.25 m for x and 13 N for restoring force in equation (1), 13 N = − k ( − 0.25 m ) k = 13 N 0.25 m k = 52 N m Substitute 52 for k , 8 for c and 2 for m in equation (3), 2 x ″ + 8 x ′ + 52 x = 0 Find the auxiliary equation using equation (4). 2 r 2 + 8 r + 52 = 0 Find the value of ω using equation (6). Substitute 52 for k , 8 for c and 2 for m in equation (6), ω = 4 ( 2 ) ( 52 ) − ( 8 ) 2 2 ( 2 ) = 352 4 = 16 × 22 4 = 4 22 4 Simplify ω as follows. ω = 22 Find the value of c 2 m . c 2 m = 8 2 ( 2 ) = 2 Substitute 22 for ω , and 2 for c 2 m in equation (7), x ( t ) = e − ( 2 ) t ( c 1 cos 22 t + c 2 sin 22 t ) (8) Since, the spring starts at equilibrium the value of x ( 0 ) = 0 (b) To determine To graph: The position function of the mass.

Bundle: Calculus: Early Transcendentals, 8th + WebAssign Printed Access Card for Stewart's Calculus: Early Transcendentals, 8th Edition, Multi-Term

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ISBN: 9781305597624

Author: James Stewart

Publisher: Cengage Learning

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Chapter 17.3, Problem 4E

(a)

To determine

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

(b)

To determine

To graph: The position function of the mass.

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The position of the mass at any time t .

Solution Summary: The author explains how to find the position of the mass at any time t.