Chapter 14, Problem 99A

To determine

Tension in the string.

Answer to Problem 99A

The tension is the string is $\text{63}\text{.7 N}$

Explanation of Solution

Given:

Length of string, $l=5.30\text{m}$

Mass of string, $m=15.0\text{g}$

Frequency, $f=125\text{Hz}$

Wavelength, $\lambda =120.0\text{cm}$

Formula Used:

Golden Rule for waves is written as,

  $v=\lambda f$

Where, $v=$ velocity, $f=$ frequency and $\lambda =$ Wavelength

Formula to calculate linear density,

  $\mu =\frac{m}{L}$

Where, $m=$ Mass of string and $l=$ length of string

Formula to calculate the speed of a wave on a string under tension:

  $v=\sqrt{\frac{F_T}{\mu }}$

Where, $F_T=$ The tension in the string and

  $\mu =$ The mass per length of the string.

Calculation:

Firstly, find the values of $v$ and $\mu$ as they are to be used in the formula of tension in the string $F_T$ .

Calculation for velocity,

Apply the formula, $v=\lambda f$ , and put the given values,

  $\text{= (1}\text{.200 m)(125 Hz) =1}{\text{.50×10}}^{\text{2}}\text{ m/s}$

Calculation for mass per length of the string,

Apply the formula, $\mu =\frac{m}{L}$ , and put the given values,

  $\text{15}\text{.0 g = 1}{\text{.50 × 10}}^{\text{-2}}\text{ kg}$

  $\begin{array}{c}\mu =\frac{m}{L}\\ =\frac{\text{1}{\text{.50 × 10}}^{\text{-2}}\text{ kg}}{5.30\text{m}}\\ =\text{2}{\text{.83×10}}^{\text{-3}}\text{ kg/m}\end{array}$

Now calculation for a tension in the string,Apply the formula, $v=\sqrt{\frac{F_T}{\mu }}$ , and put the given values. This formula is also written as, $F_T=v^2\mu$

  $\begin{array}{l}\text{= (1}{\text{.50×10}}^{\text{2}}{\text{ m/s)}}^{\text{2}}\text{(2}{\text{.83×10}}^{\text{-3}}\text{ kg/m) }\\ \text{= 63}\text{.7 N}\end{array}$

Conclusion:

Hence the tension in the string to make the wavelength of wave $\text{1}\text{.200 m}$ with frequency of $\text{125 Hz}$ is $\text{63}\text{.7 N}$ .