Q. 42 An object of mass m = 40 grams a... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 8: Q. 42 (page 555)

An object of mass m = 40 grams attached to a coiled spring with damping factor b = 0.75 gram/second is pulled down a distance a = 15 centimeters from its rest position and then released. Assume that the positive direction of the motion is up and the period is T = 5 seconds under simple harmonic motion.
(a) Develop a model that relates the distance d of the object from its rest position after t seconds.
*(b) Graph the equation found in part (a) for 5 oscillations.

Short Answer

Expert verified

The equation is $d (t) = 15 e^{- 0.75 t / 80} \cos (\sqrt{{\frac{4 蟺}{25}}^{2} - \frac{{0.75}^{2}}{160^{2}}} t)$

The graph of the motion is

Step by step solution

Part (a) Step 1. Given Information

The mass of the object is 40 grams.
Distance from rest position is 15 centimeters.
Damping Factor is $0.75 g r a m / s e c$ .
The time period is 5 seconds.

Part (a) Step 2. Develop the model

It is given that $m = 40, a = 15, b = 0.75, t = 5$ .
As the period is 5 seconds, so, $\begin{array}{rcl} \frac{2 蟺}{蝇} & = & 5 \\ 蝇 & = & \frac{2 蟺}{5} \end{array}$
Substitute the values into the equation $d (t) = a e^{- b t / 2 m} \cos (\sqrt{蝇^{2} - \frac{b^{2}}{4 m^{2}}} t)$ and simplify.

$\begin{array}{rcl} d (t) & = & 15 e^{- 0.75 t / 80} \cos (\sqrt{{(\frac{2 蟺}{5})}^{2} - \frac{{0.75}^{2}}{160^{2}}} t) \\ = & 15 e^{- 0.75 t / 80} \cos (\sqrt{{\frac{4 蟺}{25}}^{2} - \frac{{0.75}^{2}}{160^{2}}} t) \end{array}$

Part (b) Step 1. Given Information

The mass of the object is 40 grams.
Distance from rest position is 15 centimeters.
Damping Factor is $0.75 g r a m / s e c$ .
The time period is 5 seconds.

Part (b) Step 2. Plot the Graph

Use the obtained equation to plot the graph of the function.

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91影视

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Develop the model

Part (b) Step 1. Given Information

Part (b) Step 2. Plot the Graph

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