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 $l_{A} = 0 .3 m$ and that B is $l_{B} = 0 .75 m .$ 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 $l_{A} = 0 .3 m$ and that B is $l_{B} = 0 .75 m .$ 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

Stationary waves

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