(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.

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.3, Problem 3E

Chapter 17.3, Problem 5E

Students have asked these similar questions

Pls help ASAP

9. a) Determie values of a and b so that the function is continuous. ax - 2b f(x) 2 x≤-2 -2x+a, x ≥2 \-ax² - bx + 1, −2 < x < 2) 9b) Consider f(x): = 2x²+x-3 x-b and determine all the values of b such that f(x) does not have a vertical asymptote. Show work.

Chapter 17 Solutions

Calculus: Early Transcendentals

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 mass at any time t .

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