Question

A heavy piece of hanging sculpture is suspended by a 90-cmlong, 5.0 g steel wire. When the wind blows hard, the wire hums at its fundamental frequency of 80 Hz. What is the mass of the sculpture?

Short answer

The mass of the sculpture is 12 kg.$12kg$.

Step-by-step solution

Given information

The length of the wire is $L=90cm=0.90m$

The mass of the steel wire is $m=5.0g=5\times {10}^{-3}kg$

The fundamental frequency is$f=80Hz$

mass of the sculpture 

For the stretched wire vibrating at its fundamental frequency, the wavelength of the standing wave is given by $\lambda =2L$.

The wave speed on the steel wire is

$$\begin{gathered} v_{wire}=f\lambda \\ =f\left(2L\right) \\ =80\times 2\times 0.90 \\ =144m/s \end{gathered}$$

The linear density is given by

$$\begin{gathered} \mu =\frac{m}{L} \\ =\frac{5\times {10}^{-3}}{0.90} \\ =5.555\times {10}^{-3}kg/m \end{gathered}$$

The wave speed is also given as role="math" localid="1650002637854" $v=\sqrt{\frac{T_s}{\mu }}$.

The tension $T_s$in the wire equals the weight of the sculpture or $Mg$. Thus,

$$\begin{gathered} v=\sqrt{\frac{Mg}{\mu }} \\ M=\frac{v^2\mu }{g} \\ =\frac{{144}^2\times 5.555\times {10}^{-3}}{9.8} \\ =12kg \end{gathered}$$

Therefore, the mass of the sculpture is $12kg$.