Two metallic strings A and B of different materials are connected in series forming a joint. The strings have similar $$cross-sectional$$ area a $$=$$ $$1$$ $$mm2$$. The length of A is $$lA=0.3m$$ and that B is $$lB=0.75m$$. One end of the combined string is tied with a support rigidly and the other end is loaded with a block of mass m $$=$$ $$25.2$$ kg passing over a frictionless pulley. Transverse waves are set up in the combined string using an external source of variable frequency, calculate the lowest frequency (in$$ $$Hz) for which standing waves are observed such that the joint is a node.The densities of A and B are $$6.3$$ x $$103$$ $$kg/m3$$ and $$2.8$$ x $$103$$ $$kg/m3$$ respectively.

Question

Two metallic strings A and B of different materials are connected in series forming a joint. The strings have similar $$cross-sectional$$ area a $$=$$ $$1$$ $$mm2$$. The length of A is $$lA=0.3m$$ and that B is $$lB=0.75m$$. One end of the combined string is tied with a support rigidly and the other end is loaded with a block of mass m $$=$$ $$25.2$$ kg passing over a frictionless pulley. Transverse waves are set up in the combined string using an external source of variable frequency, calculate the lowest frequency (in$$ $$Hz) for which standing waves are observed such that the joint is a node.The densities of A and B are $$6.3$$ x $$103$$ $$kg/m3$$ and $$2.8$$ x $$103$$ $$kg/m3$$ respectively.

Infinity Learn · Accepted Answer

Let p $$=$$ number of loops formed in string A and q $$=$$ number of loops formed in string BBoth p and q will be integer numbers, as the one end is fixed rigidly, the other end is loaded and the joint P is a node.For string A, $$nA=p2lAT$$μ$$A=p2lATa$$ρAFor string B, $$nB=q2lBT$$μ$$B=q2lBTa$$ρBAs both the strings are to be exerted by the same source of variable frequency, hence $$nA=nB$$.          $$p2lATa$$ρ$$A=q2lBTa$$ρB          $$pq=lAlB$$ρAρ$$B=0.30.756.3$$×$$1032.8$$×$$103$$           $$=2594=25$$×$$32=35nmin=32lATa$$ρ$$A=32$$×$$0.325.2$$×$$1010$$−$$6$$×$$6.3$$×$$103=1000$$ Hz

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