Chapter 4.2, Problem 76E

a.

To determine

To find: what is the initial displacement ( t =0)

Answer to Problem 76E

  $y(0)=0.25$ .

Explanation of Solution

Given information: The displacement from equilibrium of an oscillating weight

suspended by a spring and subject to the damping effect of friction is given by,$y(t)=\frac{1}{4}{e}^{-t}\cos 6t.$

Calculation:

  $\begin{array}{l}y(t)=\frac{1}{4}{e}^{-t}\cos 6t.\\ y(0)=\frac{1}{4}{e}^{-0}\cos 6(0)=\frac{1}{4}\times 1\times 1=\mathrm{0.25.}\\ y(0)=0.25\end{array}$

b.

To determine

To use: a graphing utility to complete the table.

t	0.50	1.02	1.54	2.07	2.59
y

Answer to Problem 76E

Explanation of Solution

Given information: The displacement from equilibrium of an oscillating weight

suspended by a spring and subject to the damping effect of friction is given by,$y(t)=\frac{1}{4}{e}^{-t}\cos 6t.$

Calculation:

By using graphing utility the completed table shown below.

t	0.50	1.02	1.54	2.07	2.59
y	-0.150	0.089	-0.052	0.031	-0.0185

c.

To determine

To find: the approximate times when the weight is at its maximum distance from equilibrium are shown in the table in part (b) and explain why the magnitude of the maximum displacement is decreasing and what causes this decrease in maximum displacement in the physical system and what factor in the model measures this decrease.

Answer to Problem 76E

The magnitude of displacement is decreasing due to the fact that the pendulum is slowing down because of the function of friction.

Explanation of Solution

Given information: The displacement from equilibrium of an oscillating weight

suspended by a spring and subject to the damping effect of friction is given by,$y(t)=\frac{1}{4}{e}^{-t}\cos 6t.$

Calculation:

The magnitude of anything is always positive, therefore ignore the negative sign in the data table of part (b) and concentrate on the values which are reducing due to friction and loss of energy.

The magnitude of displacement is decreasing due to the fact that the pendulum is slowing down because of the function of friction

d.

To determine

To find: the first two times when the weight is at the equilibrium point ( y =0).

Answer to Problem 76E

x =0.262 s , x=0. 785s, the pendulum’s weight is at equilibrium point ( y =0).

Explanation of Solution

Given information: The displacement from equilibrium of an oscillating weight

suspended by a spring and subject to the damping effect of friction is given by,$y(t)=\frac{1}{4}{e}^{-t}\cos 6t.$

Calculation:

Find the zero’s that is where the graph of the function crosses x axis for first two times after 0 in the graph. This is happened at,

x =0.262 s , x=0. 785s, the pendulum’s weight is at equilibrium point ( y =0).