Question

an object of mass m (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in 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 using a graphing utility.

Given values: m = 10, a = 5, b = 0.8, T = 3

Short answer

a)the equation can be given as

$d\left(t\right)=5e^{-0.04t}\cos (\sqrt{\left(\frac{4{\pi }^2}{9}\right)-\left(\frac{0.64}{400}\right)}t$

b) plot the graph

Step-by-step solution

Given information

We are given values m = 10, a = 5, b = 0.8, T = 3

Part a) Step 1: Find angular velocity

We get,

$$\begin{gathered} \omega =\frac{2\pi }{T} \\ \omega =\frac{2\pi }{3} \end{gathered}$$

Part a) Step 2: Find the equation

We get,

$$\begin{gathered} d\left(t\right)=a{e}^{\frac{-bt}{2m}}\cos \left(\sqrt{{\omega }^2-\frac{b^2}{4m^2}}t\right) \\ d\left(t\right)=5e^{\frac{-0.8t}{2\left(10\right)}}\cos \left(\sqrt{{\left(\frac{2\pi }{3}\right)}^2-\frac{0.8^2}{4{\left(10\right)}^2}}t\right) \\ d\left(t\right)=5e^{-0.04t}\cos \left(\sqrt{\left(\frac{4{\pi }^2}{9}\right)-\frac{0.64}{400}}t\right) \end{gathered}$$

Plot the graph

We get,