Question

When a massive aluminum sculpture is hung from a steel wire, the fundamental frequency for transverse standing waves on the wire is 213.0 Hz . The sculpture (but not the wire) is then completely submerged in water.

What is the new fundamental frequency? (Hint: The density of the water is 1000 kg/m3, and the density of the aluminium is 2700 kg/m3.)

Express your answer using two significant figures.

f = __________ Hz

Answer 1

The fundamental frequency , f  = 213 Hz

Density of water , ρ_w = 1000 kg /m^3

Density of aluminium , ρ_al = 2700 kg /m^3

The formula for fundamental frequency is

                   f = (1 /2L) √T / μ

Here , frequency (f) ∞ √Tension (T)

f _air / f_water = √T_air / √T_water      -------(1)

Tension in the wire , when the sculpture in air is

T_air = ρ_al g V_al

T_air = (2700 kg /m^3)(9.8 m/s^2) V

T_air = 26460V N

According to Archemedes principle, the net force acting

on the object dubmerged in the water is

T' + F_B - W = 0   

T' = ρ_al g V_al -  ρ_w g V_al

  T' = 26460*V - 1000kg/m^3*9.8m/s^2*V

T' = 16660*V

put into (1)

213/fwater = sqrt(26460/16660)

fwater = 189.74 Hz