Chapter 5.6, Problem 27E

A Bobbing Cork A cork floating in a lake is bobbing in simple harmonic motion. Its displacement above the bottom of the lake is modeled by

$y=0.2\cos 20\pi t+8$

where y is measured in meters and t is measured in minutes.

1. (a) Find the frequency of the motion of the cork.
2. (b) Sketch a graph of y.
3. (c) Find the maximum displacement of the cork above the lake bottom.

To determine

The frequency for the motion of the cork floating in a lake.

Answer to Problem 27E

The frequency of a cork in simple harmonic motion is 10 cycles per minute.

Explanation of Solution

Given:

The equation for the displacement above the bottom of the lake is,

$y=0.2\cos 20\pi t+8$

Where, y is in meters and t is in minutes.

Definition:

The equation for the simple harmonic motion which describes the displacement y of an object at time t is,

$y=A\sin \left(kt-b\right)$

Where, A is the amplitude, b is the phase and k is the constant.

Calculation:

The equation for the displacement of the cork is,

$y=0.2\cos 20\pi t+8$ (1)

The general equation for the displacement is,

$y=A\sin \left(kt-b\right)$ (2)

Compare both the equation (1) and (2), the value of k is $20\pi$.

The formula to calculate frequency of the cork is,

$f=\frac{k}{2\pi }$ (3)

Substitute the value $20\pi$ for $k$ in equation (3),

$\begin{array}{c}f=\frac{20\pi }{2\pi }\\ =10\end{array}$

Thus, the frequency of the cork is 10 cycles per minute.

To determine

To sketch: The graph for the cosine curve $y=0.2\cos 20\pi t+8$.

Explanation of Solution

Sketch the graph for the cosine curve $y=0.2\cos 20\pi t+8$,

Figure (1)

From the Figure (1), it is showed that graph is the cosine curve with oscillation between positive amplitude $7.8$ to $8.2$.

To determine

The maximum displacement of the cork above the bottom of the lake.

Answer to Problem 27E

The maximum displacement for the cork is $8.2\text{m}$.

Explanation of Solution

The amplitude of the cork is 0.2.

The equation for the displacement of the cork is,

$y=0.2\cos \left(20\pi t+8\right)$ (4)

For the maximum displacement of the cork, the cosine function must be maximum,

So, the cosine function is maximum at $t=0$.

Substitute 0 for t in equation (3),

$y=0.2\cos 20\pi \cdot 0+8$

Solve the equation,

$\begin{array}{c}y=0.2\cos 0+8\left[\therefore \cos 0=1\right]\\ =0.2+8\\ =8.2\text{m}\end{array}$

Thus, the value of maximum displacement is $8.2\text{m}$.