Chapter 17.3, Problem 8E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A spring has a mass of 1 kg and its damping constant is *c* = 10. The spring starts from its equilibrium position with a velocity of 1 m/s. Graph the position function for the following values of the spring constant *k*: 10, 20, 25, 30, 40. 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 real roots.

Write the expression for condition of over 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

Consider

Substitute

Substitute

Substitute

Differentiate equation (5) with respect to

Substitute

Substitute

Solve for

Simplify equation as follows.

Substitute

Substitute

Substitute different values for

Table 1

0 | 0 |

0.2 | 0.078179 |

0.4 | 0.073906 |

0.6 | 0.059579 |

0.8 | 0.046596 |

1 | 0.036209 |

1.2 | 0.028098 |

1.4 | 0.021797 |

1.5 | 0.019198 |

2 | 0.010174 |

2.5 | 0.005391 |

3 | 0.002857 |

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

Consider

Substitute

Substitute

Substitute

Differentiate equation (9) with respect to

Substitute

Substitute

Solve for

Simplify equation as follows.

Substitute

Substitute

Substitute different values for

Table 2

0 | 0 |

0.2 | 0.075884 |

0.4 | 0.061565 |

0.6 | 0.039658 |

0.8 | 0.023826 |

1 | 0.013952 |

1.2 | 0.008089 |

1.4 | 0.00467 |

1.5 | 0.003546 |

2 | 0.000893 |

2.5 | 0.000225 |

3 |

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

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