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The speed v of a transverse wave travelling on a string depends on tension force (T) and mass per unit length. Using Dimensional method derived the formula of speed?
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The speed v of a transverse wave travelling on a string depends on ten...
Let the tension in the string be F and mass per unit length be u.
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The speed v of a transverse wave travelling on a string depends on ten...
Derivation of the Formula for Speed of a Transverse Wave
- Given Parameters:
Let the speed of the transverse wave be represented by v, tension force by T, and mass per unit length by μ.
- Formula:
The speed of a transverse wave is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
- Using Dimensional Analysis:
1. Write the dimensions of each parameter:
\[ [v] = LT^{-1} \]
\[ [T] = MLT^{-2} \]
\[ [\mu] = ML^{-1} \]
2. Substitute the dimensions into the formula:
\[ LT^{-1} = \sqrt{\frac{MLT^{-2}}{ML^{-1}}} \]
3. Simplify the equation:
\[ LT^{-1} = \sqrt{\frac{LT^{-2}}{L^{-1}}} \]
\[ LT^{-1} = \sqrt{L^2T^{-2}} \]
\[ LT^{-1} = LT^{-1} \]
- Conclusion:
By using dimensional analysis, we have derived the formula for the speed of a transverse wave:
\[ v = \sqrt{\frac{T}{\mu}} \]
This formula shows that the speed of a transverse wave is directly proportional to the square root of the tension force and inversely proportional to the square root of the mass per unit length of the string.
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The speed v of a transverse wave travelling on a string depends on tension force (T) and mass per unit length. Using Dimensional method derived the formula of speed?
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