A rope of 3.2 m is being applied with a tensile force of 180 N and maintains a transverse wave speed of 190 m/s. Which mass does the rope have?

To determine the mass of the rope, we can use the relationship between the speed of a wave on a string, the tension in the string, and the linear mass density of the string.

The wave speed \( v \) on a string is given by:

\[
v = \sqrt{\frac{T}{\mu}}
\]

Where:
- \( v \) is the wave speed (190 m/s),
- \( T \) is the tension in the rope (180 N),
- \( \mu \) is the linear mass density (mass per unit length) of the rope.

First, solve for \( \mu \):

\[
\mu = \frac{T}{v^2}
\]

Substitute the given values:

\[
\mu = \frac{180\ \text{N}}{(190\ \text{m/s})^2}
\]

\[
\mu = \frac{180}{36100}\ \text{kg/m}
\]

\[
\mu \approx 0.004986\ \text{kg/m}
\]

Next, the mass \( m \) of the rope is calculated using the linear mass density \( \mu \) and the length \( L \) of the rope:

\[
m = \mu \times L
\]

Substitute the values:

\[
m = 0.004986\ \text{kg/m} \times 3.2\ \text{m}
\]

\[
m \approx 0.0156\ \text{kg}
\]

So, the mass of the rope is approximately **0.0156 kg**.

Would you like more details or have any questions? Here are some related questions to consider:

1. How would the mass of the rope change if the wave speed was doubled?
2. What would happen to the wave speed if the tension in the rope was increased by 50%?
3. How does the length of the rope affect the wave speed if the tension and mass are constant?
4. Can you calculate the tension required for a wave speed of 250 m/s on the same rope?
5. What is the significance of linear mass density in understanding wave propagation on a string?

**Tip:** Remember that wave speed on a string is directly related to the square root of the tension, making it sensitive to changes in tension.

Discover how to calculate the mass of a 3.2 m rope subjected to a 180 N tensile force, maintaining a wave speed of 190 m/s. This problem explores wave mechanics, including the relationship between wave speed, tension, and linear mass density, and is suitable for students in Grades 10-12.

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