Q16E - University Physics with Modern Physics

Question

With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?

Step-by-Step Solution

Answer

The tension with which the rope should be stretched is 43.2N.

Step 1:Determination of the wave speed on a string and wave speed formula

The speed of a wave is the distance travelled by a given point on a wave ina given interval of time.
 

Thespeedof a wave on a string in terms of the tension T and the mass per unit length $\mu$ is given by $v=\sqrt{\frac{T}{\mu }}$
 
And the general speed of a wave in terms of the wavelength $\lambda$ and the frequency f is given by:

$v=f\times \lambda$ 

Step 2: Calculation using the wave speed on a string and wave speed formula

The length of the rope is l = 2.50m, its mass is m = 0.120kg, the frequency of the wave in the rope is f = 40Hz and its wavelength is $\lambda$= 0.750m. 
 
 First, calculate the wave speed by substituting for $\lambda$ and f into the wave speed formula:

$$\begin{gathered} v=\left(40s^{-1}\right)\times \left(0.750m\right) \\ =30m/s \end{gathered}$$

 Then calculate the linear mass density of the rope:

$$\begin{gathered} \mu =\frac{m}{l} \\ =\frac{0.12kg}{2.50m} \\ =0.048kg/m \end{gathered}$$

 
 Finally,put in the values for v and f into the wave speed on a string formula and solve for the value of tension on the rope:

 

$$\begin{gathered} 30m/s=\sqrt{\frac{T}{0.048kg/m}} \\ T=0.048\times {\left(30\right)}^2 \\ =43.2N \end{gathered}$$ 

 

 Therefore, the required tension on the rope is, $T=43.2N$ .