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Standard XII

Physics

Dimensional Analysis

Show dimensio...

Question

Show dimensionally that the frequency (a) of transverse wanes in a string of length / and mass per unit length $m$ under a tension T is given by, $n = k \sqrt{T} / l \sqrt{m}$

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Solution

Let $n = k T^{a} l^{b} m^{c}$ where $k$ =dimensionless constant

$[T] = M L T^{- 2}$
$[l] = L$
$[m] = M L^{- 1}$

And $[n] = T^{- 1}$

$⟹ T^{- 1} = (M L T^{- 2})^{a} L^{b} (M L^{- 1})^{c} = M^{(a + c)} L^{(a + b - c)} T^{- 2 a}$

$⟹ - 2 a = - 1$ and $a + c = 0$ and $a + b - c = 0$

Solving we get-

$a = \frac{1}{2}, c = \frac{- 1}{2}$ and $b = - 1$

$⟹ n = k \frac{\sqrt{T}}{l \sqrt{m}}$

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Show dimensionally that the frequency (a) of transverse wanes in a string of length / and mass per unit length m under a tension T is given by, n=k√T/l√m

Let n=kTalbmc where k=dimensionless constant[T]=MLT−2[l]=L[m]=ML−1And [n]=T−1⟹T−1=(MLT−2)aLb(ML−1)c=M(a+c)L(a+b−c)T−2a⟹−2a=−1 and a+c=0 and a+b−c=0Solving we get-a=12,c=−12 and b=−1⟹n=k√Tl√m

Show dimensionally that the frequency (a) of transverse wanes in a string of length / and mass per unit length $m$ under a tension T is given by, $n = k \sqrt{T} / l \sqrt{m}$