Chapter 14, Problem 96A

(a)

To determine

The original period of the ride.

Explanation of Solution

Given:

The originalnumber of swings, $n=8$ ,

The time taken is, $t=1\min =60\text{ s }$

The new number of swings, $n\text{'}=6$ ,

Formula used:

The expression for frequency is given by,

  $f=\frac{n}{t}$     .......(1)

Here, $n$ original number of swings $t$ is the time, $f$ is the frequency,

Calculation:

(a) Substituting $8$ in $n$ and $60\text{ s}$ in $t$ in equation (1),

  $\begin{array}{l}f=\frac{8}{60}\\ f=0.133\text{ Hz}\end{array}$

Conclusion:

Hence, the required timeis $7.518\text{ s}$ .

(b)

To determine

The new period of the ride.

Explanation of Solution

Given:

The original number of swings, $n=8$ ,

The time taken is, $t=1\min =60\text{ s }$

The new number of swings, $n\text{'}=6$ ,

Formula used:

The relation between time period and frequency is give by,

  $T=\frac{1}{f}$    .......(1)

Here, $f$ is the frequency, $T$ is the time period of the $l$ is the length of the carriage.

Calculation:

Substituting 0.133 Hz in f in equation (1),

  $\begin{array}{c}T=\frac{1}{0.133}\\ =7.518\text{ s}\end{array}$

Conclusion:

Hence, the required time period is $7.518\text{s}\text{.}$

(c)

To determine

The new frequency.

Explanation of Solution

Given:

The original number of swings, $n=8$ ,

The time taken is, $t=1\min =60\text{ s }$

The new number of swings, $n\text{'}=6$ ,

Formula used:

The relation between frequency and time period is give by,

  $f=\frac{1}{T}$     .......(1)

Here, $f$ is the frequency, $T$ is the time period of the $l$ is the length of the carriage.

Calculation:

Substituting 10 s in f in equation (1)

  $\begin{array}{c}f=\frac{1}{10}\\ =0.1\text{ Hz}\end{array}$

Conclusion:

Hence, the required frequency is $0.1\text{Hz}\text{.}$

(d)

To determine

The length of the arm supporting the carriage on the larger ride.

Explanation of Solution

Given:

The original number of swings, $n=8$ ,

The time taken is, $t=1\min =60\text{ s }$

The new number of swings, $n\text{'}=6$ ,

Formula used:

  $l=g\frac{T^2}{4{\pi }^2}$    .......(1)

Here, $n$ original number of swings, $t$ is the time, $f$ is the frequency, $T$ is the time period of the $l$ is the length of the carriage and $g$ is the acceleration due to gravity.

Calculation:

For the original length of the arm, substituting 7.518 s for Tand 9.8 m/s² for g equation (1)

  $\begin{array}{l}l=9.8\frac{{7.518}^2}{4{\pi }^2}\\ l=14\text{ m}\end{array}$

For the new length of the arm, substituting 1 s for T and 9.8 m/s² for g equation (1)

  $\begin{array}{l}l=9.8\frac{{10}^2}{4{\pi }^2}\\ l=25\text{ m}\end{array}$

The length of the arm on the new structure is longer by 11m.

Conclusion:

Hence, the required length is $11\text{m}$ .

(e)

To determine

The point at which the KE is minimum.

Explanation of Solution

Given:

The original number of swings, $n=8$ ,

The time taken is, $t=1\min =60\text{ s }$

The new number of swings, $n\text{'}=6$ ,

Calculation:

The percentage increase in the length of the pendulum if the period of ride is doubled is 300% increase. If the period is doubled then the length is 4times increased because of squared relation of length and time period. Hence there would be 300% increase.

Conclusion:

Hence there would be 300% increase.