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smashd
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Question: A copper wire and steel wire with identical diameters are placed under identical tensions. The frequency of the third resonant mode for the copper wire is found to be the same as the frequency of the fourth resonant mode for the steel wire. If the length of the copper wire is 3.44 m and the density of steel is 7.85 g/cc, find the length of the steel wire.
Known Values: ## ~ ~ ~ ~ ~ L_c = 3.44 ~ m ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ ρ_s = 7.85 ~ g/cc ##
Relevant Equations: ## ~ ~ ~ ~ ~ ƒ_n = \frac{v}{λ_n} ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ λ_n = \frac{2L}{n} ~ ~ ~ ~ ~ ~ , ~ ~ ~ ~ v = \sqrt{\frac{T}{μ}} ##
Attempt at a Solution:
[tex] ƒ_{3-c} = ƒ_{4-s} [/tex]
[tex] \frac{v_c}{λ_{3-c}} = \frac{v_s}{λ_{4-s}} [/tex]
[tex] \frac{\sqrt{\frac{T}{μ_c}}}{λ_{3-c}} = \frac{\sqrt{\frac{T}{μ_s}}}{λ_{4-s}} [/tex]
[tex] \frac{\sqrt{\frac{T}{ρ_c ~ π ~ r^2}}}{\frac{2}{3} ~ L_c} = \frac{\sqrt{\frac{T}{ρ_s ~ π ~ r^2}}}{\frac{1}{2} ~ L_s} [/tex]
[tex] \frac{\frac{T}{ρ_c ~ π ~ r^2}}{\frac{4}{9} ~ L_{c}^2} = \frac{\frac{T}{ρ_s ~ π ~ r^2}}{\frac{1}{4} ~ L_{s}^2} [/tex]
[tex] \frac{T}{\frac{4}{9} ~ ρ_c ~ π ~ r^2 ~ L_{c}^2} = \frac{T}{\frac{1}{4} ~ ρ_s ~ π ~ r^2 ~ L_{s}^2} [/tex]
[tex] \frac{4}{9} ~ ρ_c ~ L_{c}^2 = \frac{1}{4} ~ ρ_s ~ L_{s}^2 [/tex]
[tex]∴ ~ L_{s} = \frac{4}{3} ~ L_{c} ~ \sqrt{\frac{ρ_c}{ρ_s}} [/tex]
My problem is that I don't know what ## ρ_c ## is. Any ideas how to solve this? Am I even going about this the right way?
Known Values: ## ~ ~ ~ ~ ~ L_c = 3.44 ~ m ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ ρ_s = 7.85 ~ g/cc ##
Relevant Equations: ## ~ ~ ~ ~ ~ ƒ_n = \frac{v}{λ_n} ~ ~ ~ ~ ~ , ~ ~ ~ ~ ~ λ_n = \frac{2L}{n} ~ ~ ~ ~ ~ ~ , ~ ~ ~ ~ v = \sqrt{\frac{T}{μ}} ##
Attempt at a Solution:
[tex] ƒ_{3-c} = ƒ_{4-s} [/tex]
[tex] \frac{v_c}{λ_{3-c}} = \frac{v_s}{λ_{4-s}} [/tex]
[tex] \frac{\sqrt{\frac{T}{μ_c}}}{λ_{3-c}} = \frac{\sqrt{\frac{T}{μ_s}}}{λ_{4-s}} [/tex]
[tex] μ_c = \frac{m_c}{L_c} ~ ~ ~ , ~ ~ ~ ρ_c = \frac{m_c}{V_c} ~ ~ ~ ⇒ ~ ~ ~ m_c = ρ_c ~ V_c = ρ_c ~ π ~ r^2 ~ L_c[/tex]
[tex] μ_c = \frac{m_s}{L_s} ~ ~ ~ , ~ ~ ~ ρ_s = \frac{m_s}{V_s} ~ ~ ~ ⇒ ~ ~ ~ m_s = ρ_s ~ V_s = ρ_s ~ π ~ r^2 ~ L_s[/tex]
[tex] μ_c = ρ_c ~ π ~ r^2 [/tex]
[tex] μ_s = ρ_s ~ π ~ r^2 [/tex]
[tex] \frac{\frac{T}{ρ_c ~ π ~ r^2}}{\frac{4}{9} ~ L_{c}^2} = \frac{\frac{T}{ρ_s ~ π ~ r^2}}{\frac{1}{4} ~ L_{s}^2} [/tex]
[tex] \frac{T}{\frac{4}{9} ~ ρ_c ~ π ~ r^2 ~ L_{c}^2} = \frac{T}{\frac{1}{4} ~ ρ_s ~ π ~ r^2 ~ L_{s}^2} [/tex]
[tex] \frac{4}{9} ~ ρ_c ~ L_{c}^2 = \frac{1}{4} ~ ρ_s ~ L_{s}^2 [/tex]
[tex]∴ ~ L_{s} = \frac{4}{3} ~ L_{c} ~ \sqrt{\frac{ρ_c}{ρ_s}} [/tex]
My problem is that I don't know what ## ρ_c ## is. Any ideas how to solve this? Am I even going about this the right way?
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