Chapter 12.3, Problem 85AYU

(a)

To determine

To find: The length of the arc of the 10^th swing.

Answer to Problem 85AYU

  $0.7\text{ feet}$

Explanation of Solution

Given: Initially, a pendulum swings through an arc of 2 feet. On each successive, the length of the arc is 0.9 of the previous length.

Initial length of the pendulum = 2 feet.

After each swing loose 0.1 length of the previous length.

After 1 swing length of arc $=0.9\times 2=1.8\text{ feet}$

After 2 bounce height of ball $=0.9\times 1.8=1.62\text{feet}$

After 10 bounce height of ball $={\left(0.9\right)}^{10}\times 2=0.7\text{feet}$

(b)

To determine

To find: The length of arc less than 1 foot.

Answer to Problem 85AYU

  $7$

Explanation of Solution

Given: Initially, a pendulum swings through an arc of 2 feet. On each successive, the length of the arc is 0.9 of the previous length.

Initial height of the ball, $a_0=30\text{ feet}$ .

After each bounces up loose one-fourth height of the previous one.

Length of arc after nth swung $l_n=2{\left(0.9\right)}^n\text{ feet}$

Length is less than 1 foot.

  $\begin{array}{l}2{\left(0.9\right)}^n<1\\ {0.9}^n<0.5\\ n<\frac{\ln 0.5}{\ln 0.9}\\ n<6.57\end{array}$

Before 7 swung length of arc less than 1 foot.

(c)

To determine

To find: The number of bounce when height is less than 6 inches.

Answer to Problem 85AYU

  $0.41\text{ foot}$

Explanation of Solution

Given: Initially, a pendulum swings through an arc of 2 feet. On each successive, the length of the arc is 0.9 of the previous length.

Initial height of the ball, $a_0=2\text{ feet}$ .

Length of arc after nth swung $l_n=2{\left(0.9\right)}^n\text{ feet}$

Put n=15

Length of arc after 15th swung $l_{15}=2{\left(0.9\right)}^{15}\text{=0}\text{.41 feet}$

(d)

To determine

To find: The total length when stops swung.

Answer to Problem 85AYU

  $38\text{ feet}$

Explanation of Solution

Given: Initially, a pendulum swings through an arc of 2 feet. On each successive, the length of the arc is 0.9 of the previous length.

Initial height of the ball, $a_0=2\text{ feet}$ .

Length of arc after nth swung $l_n=2{\left(0.9\right)}^n\text{ feet}$

Total length $,S_n=l_0+2l_1+2l_2+\mathrm{.............}\infty$

  $\begin{array}{l}{S}_n=2+2\left(0.9\times 2+{\left(0.9\right)}^2\times 2+\mathrm{.........}\right)\\ S_n=2+2\times 0.9\times \frac{2}{1-0.9}\\ S_n=38\text{ feet}\end{array}$

Hence, total length of arc before its stops is $38\text{ feet}$ .