Math Problem Statement
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?
Solution
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 on a string is given by:
Where:
- is the wave speed (190 m/s),
- is the tension in the rope (180 N),
- is the linear mass density (mass per unit length) of the rope.
First, solve for :
Substitute the given values:
Next, the mass of the rope is calculated using the linear mass density and the length of the rope:
Substitute the values:
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:
- How would the mass of the rope change if the wave speed was doubled?
- What would happen to the wave speed if the tension in the rope was increased by 50%?
- How does the length of the rope affect the wave speed if the tension and mass are constant?
- Can you calculate the tension required for a wave speed of 250 m/s on the same rope?
- 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.
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Math Problem Analysis
Mathematical Concepts
Wave Mechanics
Linear Mass Density
Formulas
v = sqrt(T / μ)
μ = T / v^2
m = μ × L
Theorems
Wave Speed Formula on a String
Suitable Grade Level
Grades 10-12
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