Two wires are fixed on a sonometer. Their tensions are in the ratio $$8$$ :$$1$$, their lengths are in the ratio $$36$$:$$35$$, the diameters are in the ratio $$4$$:$$1$$ and densities are in the ratio $$1$$:$$2$$. If the value of higher frequency is $$36Hz$$, then lower frequency is p×qHz, here both p and q are integers. The value of $$p-q$$ is

Question

Two wires are fixed on a sonometer. Their tensions are in the ratio $$8$$ :$$1$$, their lengths are in the ratio $$36$$:$$35$$, the diameters are in the ratio $$4$$:$$1$$ and densities are in the ratio $$1$$:$$2$$. If the value of higher frequency is $$36Hz$$, then lower frequency is p×qHz, here both p and q are integers. The value of $$p-q$$ is

Infinity Learn · Accepted Answer

Given,          $$T1T2=81$$,$$L1L2=3635D1D2=41$$,ρ$$1$$ρ$$2=12Let$$ μ$$1$$ and μ$$2$$ be the linear mass densities, then∴ μ$$1=$$π×$$D124$$×ρ$$1$$ and μ$$2=$$π×$$D224$$×ρ$$2$$            $$($$μ$$=$$ρ$$S)$$∴ μ$$1$$μ$$2=D1D22$$×ρ$$1$$ρ$$2=412$$×$$12=81$$∴ $$f1f2=L2L1$$×$$T1T2$$×μ$$2$$μ$$1=353681$$×$$18=3536$$           $$f=12LT$$μWe have                 $$f2$$>$$f1f2=36$$∴$$f1=35=7$$×$$5p=7$$ ; $$q=5$$ ⇒$$p-q=2$$

Column-I		Column-II
(a)	$y = Asin \frac{πx}{L} \sin ωt$	(p)	1 antinode, 2 nodes
(b)	$y = Acos \frac{πx}{L} \sin ωt$	(q)	3 nodes, 2 antinodes
(c)	$y = Asin \frac{2 πx}{L} \sin ωt$	(r)	2 antinodes, 1 node
(d)	$y = Acos \frac{2 πx}{L} \sin ωt$	(s)	3 antinodes, 2 nodes

Stationary waves

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Stationary waves

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Given, T1T2=81,L1L2=3635D1D2=41,ρ1ρ2=12Let μ1 and μ2 be the linear mass densities, then∴ μ1=π×D124×ρ1 and μ2=π×D224×ρ2 (μ=ρS)∴ μ1μ2=D1D22×ρ1ρ2=412×12=81∴ f1f2=L2L1×T1T2×μ2μ1=353681×18=3536 f=12LTμWe have f2>f1f2=36∴f1=35=7×5p=7 ; q=5 ⇒p-q=2

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