Q. 83 - Calculus

Question

Consider a mass hanging from the ceiling at the end of a spring. If you pull down on the mass and let go, it will oscillate up and down according to the equation

$s\left(t\right)=A\sin \left(\sqrt{\frac{k}{m}}t\right)+B\cos \left(\sqrt{\frac{k}{m}}t\right)$

where $s\left(t\right)$ is the distance of the mass from its equilibrium position, m is the mass of the bob on the end of the spring, and k is a "spring coefficient" that measures how tight or stiff the spring is. The constants A and B depend on initial conditions - specifically, how far you pull down the mass $\left(s_0\right)$ and the velocity at which you release the mass $\left(v_0\right)$. This equation does not take into effect any friction due to air resistance.

1. Determine whether or not the limit of $s\left(t\right)$ as $t\to \infty$ exists. What does this say about the long-term behavior of the mass on the end of the spring?
2. Explain how this limit relates to the fact that the equation for $s\left(t\right)$ does not take friction due to air resistance into account.
3. Suppose the bob at the end of the spring has a mass of $2$ grams and that the coefficient for the spring is $k=9$. Suppose also that the spring is released in such a way that $A=\sqrt{2}$ and $B=2$. Use a graphing utility to graph the function $s\left(t\right)$ that describes the distance of the mass from its equilibrium position. Use your graph to support your answer to part (a).

Step-by-Step Solution

Answer

Part(a) The limit does not exist as both sine and cosine oscillate between $1$ and $-1$ as $x\to \infty$.

Part(b) There is no friction to damp the oscillations of the spring as the limit does not actually exist. 

Part(c) The graph is as follows, 

Part(a) Step 1. Given Information

We are given an equation and a figure,

$s\left(t\right)=A\sin \left(\sqrt{\frac{k}{m}}t\right)+B\cos \left(\sqrt{\frac{k}{m}}t\right)$

Part(a) Step 2. Finding the limit

The limit is given by,

$$\begin{gathered} \underset{t\to \infty }{\lim }s\left(t\right)=\underset{t\to \infty }{\lim }\left[A\sin \left(\sqrt{\frac{k}{m}t}\right)+B\cos \left(\sqrt{\frac{k}{m}t}\right)\right] \\ =\underset{t\to \infty }{\lim }\left[A\sin \left(\sqrt{\frac{k}{m}}t\right)\right]+\underset{t\to \infty }{\lim }\left[B\cos \left(\sqrt{\frac{k}{m}}t\right)\right] \\ =A\sin \left(\sqrt{\frac{k}{m}}\right)+B\cos \left(\sqrt{\frac{k}{m}}\infty \right) \\ =A\sin \infty +B\cos \infty \\ =\text{ Undefined } \end{gathered}$$

The limit does not exist, since both sine and cosine oscillate between $-1$ and $1$ as $x\to \infty$.

Part(b) Step 1. About the friction due to air resistance

Since the limit does not actually exist, there is no friction to damp the oscillations of the spring.

Part(c) Step 1. Graphing the function

Put

 $$\begin{gathered} m=2, \\ k=9, \\ A=\sqrt{2},B=2 \end{gathered}$$ 

 in the equation to get as below:

$$\begin{gathered} s\left(t\right)=\sqrt{2}\sin \left(\sqrt{\frac{9}{2}t}\right)+2\cos \left(\sqrt{\frac{9}{2}}t\right) \\ =\sqrt{2}\sin \left(\frac{{\displaystyle 3}}{{\displaystyle \sqrt{2}}}t\right)+2\cos \left(\frac{{\displaystyle 3}}{{\displaystyle \sqrt{2}}}t\right) \end{gathered}$$

The graph is as follows,