Question

A spherical ball contracts in volume by $0.01\%$ when subjected to normal uniform pressure $100atm$. The Bulk modulus of its material is

Answer 1

Step 1: Given data

1. The contraction of volume is $△V=0.01$.
2. The exerted pressure is $P=100atm.$

Step 2: Bulk modulus

1. Bulk modulus is the ratio of pressure and volume strain.
2. Bulk modulus is defined by the form, $B=\frac{P}{{\displaystyle \frac{∆V}{V}}}$, where, $\mathrm{P}$ is the pressure exerted on volume $\mathrm{V}$, $\frac{∆V}{V}$ is the strain of the volume due to compression.

Step 3: Finding the bulk modulus

As we know, volume strain, $\frac{∆V}{V}$

So,

$$\begin{gathered} \frac{∆V}{V}=\frac{0.01}{100}=1\times {10}^{-4} \\ or\frac{∆V}{V}=1\times {10}^{-4} \end{gathered}$$

Now, the Bulk modulus is

$$\begin{gathered} B=\frac{P}{{\displaystyle \frac{∆V}{V}}}=\frac{100atm}{1\times {10}^{-4}} \\ B=\frac{100\times 1.01\times {10}^5}{1\times {10}^{-4}}=1.01\times {10}^{11}N{m}^{-2}\left(Since,1atm=1.01\times {10}^{5}N{m}^{-2}\right) \\ B=1.01\times {10}^{11}N{m}^{-2} \end{gathered}$$

Therefore, the bulk modulus of its material is $1.01\times {10}^{11}N{m}^{-2}$.