Chapter 17.3, Problem 7E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A spring has a mass of 1 kg and its spring constant is *k =* 100. The spring is released at a point 0.1 m above its equilibrium position. Graph the position function for the following values of the damping constant *c*: 10, 15, 20, 25, 30. What type of damping occurs in each case?

To determine

**To graph:** The position function for spring constants

Explanation

**For spring constant**
**:**

Write the expression for differential equation of motion of a spring with damping constant.

Write the expression for auxiliary equation.

Write the expression for general solution with complex roots.

Write the expression for

Write the expression for general solution with real roots.

Write the expression for condition of under damping.

Find the expression for differential equation using equation (1).

Substitute
*m*,

Find the auxiliary equation using equation (2).

Substitute
*m*,

Find the value of

Simplify equation as follows.

Compare equation (4) and (7).

Substitute

Substitute

Substitute

Find the value of

Differentiate

Differentiate equation (8) with respect to

Substitute

Substitute

Substitute

Simplify equation as follows.

Substitute

Substitute different values for

Table 1

0 | â€“0.1 |

0.2 | â€“0.01506 |

0.4 | 0.015312 |

0.6 | 0.000229 |

0.8 | â€“0.0021 |

1 | 0.000217 |

1.2 | 0.000258 |

1.4 | â€“6Eâ€“05 |

1.5 | â€“6.4Eâ€“05 |

Draw the graph using values tabulated in Table 1 as shown in Figure 1.

Find the value of

The value of

Thus, the position function for spring constant

**For spring constant**
**:**

Find the expression for differential equation using equation (1).

Substitute
*m*,

Find the auxiliary equation using equation (2).

Substitute
*m*,

Find the value of

Simplify equation as follows.

Compare equation (4) and (10).

Substitute

Substitute

Substitute

Differentiate

Differentiate equation (11) with respect to

Substitute

Substitute

Substitute

Simplify equation as follows.

Substitute

Substitute different values for

Table 2

0 | 0 |

0.2 | â€“0.01626 |

0.4 | 0.002842 |

0.6 | â€“0.00189 |

0.8 | 7Eâ€“05 |

1 | â€“9.1Eâ€“05 |

1.2 | â€“1.2Eâ€“05 |

1.4 | â€“2.6Eâ€“06 |

1.5 | â€“3.1Eâ€“06 |

Draw the graph using values tabulated in Table 2 as shown in Figure 2.

Find the value of

The value of

Thus, the position function for spring constant

**For spring constant**
**:**

Write the expression for general solution with same roots.

Write the expression for condition of critical damping.

Find the expression for differential equation using equation (1).

Substitute
*m*,

Find the auxiliary equation using equation (2).

Substitute
*m*,

Find the value of

Substitute

Substitute

Substitute

Sect-17.1 P-1ESect-17.1 P-2ESect-17.1 P-3ESect-17.1 P-4ESect-17.1 P-5ESect-17.1 P-6ESect-17.1 P-7ESect-17.1 P-8ESect-17.1 P-9ESect-17.1 P-10E

Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

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