A steel wire of negligible mass is placed under tension by attaching its bottom

Question

A steel wire of negligible mass is placed under tension by attaching its bottom to the tip of a solid copper cone of height H. The top of the wire is suspended from the ceiling. When the wire is struck, it vibrates with a fundamental frequency (n=1) of 300 Hz. If the cone is now submerged in benzene up to half its height, what is the wire's new fundamental frequency? A steel wire of negligible mass is placed under tension by attaching its bottom to the tip of a solid copper cone of height H. The top of the wire is suspended from the ceiling. When the wire is struck, it vibrates with a fundamental frequency (n=1) of 300 Hz. If the cone is now submerged in benzene up to half its height, what is the wire's new fundamental frequency?

Explanation / Answer

v = sqrt(T/u)
u = linear density = constant

f = v/2L

Volume V1 = (1/3)*pi*R^2*H

T1 = rho_c*V1*g

v1 = sqrt(T1/u)

f1 = v1/ 2L

if H' = H/2

R' = R/2

volume immersed

V2 = V1 - (1/3)*pi*R'^2*H' = 7V1/8

T2 = T1-rho_b*g*(H/2)= rho_c*V1*g - rho_b*g*(7V1/8)


T2 = (rho_c - rho_b)*(1/8)*V1*g

T2/T1 = (rho_c - rho_b)/(8*rho_c)

v2 = sqrt(T2/u)

f2 = v2/2L


f2/f1 = sqrt(T2/T1)

f2 = f1*sqrt((rho_c - rho_b)/(8*rho_c))

f2 = 300*sqrt((8.9-0.876)/(8.9))

f2 = 284.85 Hz

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