Chapter 15, Problem 1SP

(a)

To determine

The mass per unit length of the rope.

Answer to Problem 1SP

The mass per unit length of the rope is $0.3\text{kg}/\text{m}$.

Explanation of Solution

Given info: The length of the rope is $8\text{m}$ , mass of the rope is $2.4\text{kg}$.

Write the equation to find the mass per unit length of the rope.

$m=\frac{M}{L}$

Here,

$m$ is the mass per unit length of the rope

$M$ is the mass of the rope

$L$ is the length of the rope

Substitute $2.4\text{kg}$ for $M$ and $8\text{m}$ for $L$ in the equation to get $m$.

$\begin{array}{c}m=\frac{2.4\text{kg}}{8\text{m}}\\ =0.3\text{kg}/\text{m}\end{array}$

Conclusion:

Therefore, the mass per unit length of the rope is $0.3\text{kg}/\text{m}$.

(b)

To determine

The speed of the waves formed in the rope.

Answer to Problem 1SP

The speed of the waves formed in the rope is $10\text{m}/\text{s}$.

Explanation of Solution

Given info: The tension in the rope is $30\text{N}$.

Write the equation to find the velocity of waves formed in a rope under tension.

$v=\sqrt{\frac{F}{m}}$

Here,

$v$ is the velocity of the waves formed in the rope

$F$ is the tension experienced by the rope

$m$ is the mass per unit length of the rope

Substitute $30\text{N}$ for $F$ and $0.3\text{kg}/\text{m}$ for $m$ in the equation to find $v$.

$\begin{array}{c}v=\sqrt{\frac{30\text{N}}{0.3\text{kg}/\text{m}}}\\ =10\text{m}/\text{s}\end{array}$

Conclusion:

Therefore, the velocity of the waves formed in the rope is $10\text{m}/\text{s}$.

(c)

To determine

The wavelength of the waves formed in the rope.

Answer to Problem 1SP

The wavelength of the waves formed in the rope is $4\text{m}$.

Explanation of Solution

Given info: Frequency of the waves in the rope is $2.5\text{Hz}$.

Write the equation to find the wavelength of the waves formed in the rope.

$\lambda =\frac{v}{f}$

Here,

$\lambda$ is the wavelength of the wave in the rope

$v$ is the velocity of wave formed in the rope

$f$ is the frequency of wave formed in the rope

Substitute $10\text{m}/\text{s}$ for $v$ and $2.5\text{Hz}$ for $f$ in the equation to get $\lambda$.

$\begin{array}{c}\lambda =\frac{10\text{m}/\text{s}}{2.5\text{Hz}}\\ =4\text{m}\end{array}$

Conclusion:

Therefore, the wavelength of the wave formed in the rope is $4\text{m}$.

(d)

To determine

The number of complete cycles of wave that will fit on the rope.

Answer to Problem 1SP

The number of complete cycles of waves that is formed in the rope is $2$.

Explanation of Solution

The length of rope used is $8\text{m}$. The wavelength of the waves formed in the rope is $4\text{m}$.

Therefore the number of wave cycles that will be completed in the rope is found by just dividing the total length by wavelength.

Write the equation to find the number of wave cycles completed.

$n=\frac{L}{\lambda }$

Here,

$n$ is the number of cycles formed in the rope

$L$ is the length of the rope

$\lambda$ is the wavelength of waves formed in the rope

Substitute $8\text{m}$ for $L$ and $4\text{m}$ for $\lambda$ in the equation to get $n$.

$\begin{array}{c}n=\frac{8\text{m}}{4\text{m}}\\ =2\end{array}$

Conclusion:

Therefore, the number of wave cycles in the rope is $2$.

(e)

To determine

The time taken by the leading edge of the waves to go and come back from the edge of the rope.

Answer to Problem 1SP

The waves take a time of $0.8\text{s}$ to reach one end of rope and come back.

Explanation of Solution

The waves need to travel a distance of $8\text{m}$ to reach the end of the rope. Each wave has a length of $4\text{m}$. Therefore two complete cycle of wave is passing through the rope before reaching the other end.

The time taken by a wave to complete one cycle of motion is called time period. We have two cycles here and so we have to multiply the time period for one cycle by two to get the time period for two cycles.

Write the equation to find the time period of two cycles of waves.

$T=2\times \frac{1}{f}$

Here,

$T$ is the time period of the waves

$f$ is the frequency of the waves

Substitute $2.5\text{Hz}$ for $f$ in the equation to find $T$.

$\begin{array}{c}T=2\times \frac{1}{2.5\text{Hz}}\\ =0.8\text{s}\end{array}$

Conclusion:

Therefore, the time taken by the waves to reach and start back again from the end of the rope is $0.8\text{s}$.