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Obtain the equation of speed of transverse wave on tensed (stretched) string. |
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Answer» Solution :The speed of transverse waves on a string is determined by two factors, (i) the linear mass DENSITY or mass PER unit length `mu` and (ii) the TENSION T. The linear mass density, `mu` of a string is the mass m of the string divided by its length l. Therefore its dimension is `[M^(1) L ^(-1)].`The tension T has the dimension of force - namely `[M^(1) L ^(1) T ^(2)].` We have to combine `mu` and T in such a way as to generate v [dimension `(LT ^(-1) )].` It can be seen that tghe ratio `T/mu` has the dimension `[ L ^(2) T ^(-2)].` `[(T)/(mu) ] = ([ M^(2) L ^(1) T ^(-2)])/( [ M ^(1) L ^(-1)]) = [L^(2) T ^(-2)]` Therefore, if v depends only on T anad `mu,` the relation between them must be, `v = C sqrt ((T)/(mu))` Here C is a dimensionless constant and constnat C is INDEED equal to unity. The speed of transverse waves on the stretched string is therefore given by. `v = sqrt ((T)/(mu))` The speed of a wave along a stretched ideal string depends only on the tension and the linear mass density of the streing and does not DEPEND on the frequency of the wave. |
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