Category : JEE Main & Advanced
The velocity of a transverse wave in a stretched string is given by \[v=\sqrt{\frac{T}{m}}\]; where T = Tension in the string; m = Linear density of string (mass per unit length).
(1) If A is the area of cross-section of the wire then \[m=\rho A\]
\[\Rightarrow \] \[v=\sqrt{\frac{T}{\rho A}}=\sqrt{\frac{S}{\rho }}\]; where S = Stress \[=\frac{T}{A}\]
(2) If string is stretched by some weight then
\[T=Mg\]
\[\Rightarrow \] \[v=\sqrt{\frac{Mg}{m}}\]
(3) If suspended weight is immersed in a liquid of density \[\sigma \] and \[\rho =\] density of material of the suspended load then
\[T=Mg\left( 1-\frac{\sigma }{\rho } \right)\]
\[\Rightarrow \] \[v=\sqrt{\frac{Mg(1-\sigma /\rho )}{m}}\]
(4) If two rigid supports of stretched string are maintained at temperature difference of \[\Delta \theta \] then due to elasticity of string.
\[T=YA\alpha \Delta \theta \]
\[\Rightarrow \]\[v=\sqrt{\frac{YA\alpha \Delta \theta }{m}}\]
\[=\sqrt{\frac{Y\alpha \Delta \theta }{d}}\]
where Y = Young's modulus of elasticity of string, A = Area of cross section of string, \[\alpha =\] Temperature coefficient of thermal expansion, d = Density of wire \[=\frac{m}{A}\]
(5) In a solid body : \[v\,=\,\sqrt{\frac{\eta }{\rho }}\]
where \[\eta =\] Modulus of rigidity; \[\rho =\]Density of the material.
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