Q11-31E - University Physics with Modern Physics

Question

The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by 0.10% of their original volume? (b) Given that the pressure in the ocean increases by 1.0 ^* 10⁴Pa for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by 0.10% ? Does it seem that bone compression is a problem she needs to be concerned with when diving?

Step-by-Step Solution

Answer

(a) 150 atm

(b) 1.5 Km

Step 1: Given information:

Bulk modulus $\beta =15Gpa$, $\frac{\Delta V}{V_0}=-0.0010$

Step 2: Concept/Formula used:

Bulk modulus is given by the ratio of pressure applied to the corresponding relative decrease in the volume of the material. 

Mathematically, it is represented as follows:

$\mathit{\beta }\mathbf{=}\frac{\mathbf{\Delta }\mathbf{p}}{\frac{\mathbf{\Delta }\mathbf{V}}{{\mathbf{V}}_{\mathbf{0}}}}$

Where, is Bulk modulus

$\mathit{\Delta }\mathit{p}$ Is change of the pressure or force applied per unit area on the material

$\mathit{\Delta }\mathit{V}$ is Change of the volume of the material due to the compression

${\mathit{V}}_{\mathbf{0}}$ is Initial volume of the material

Step 3: Raised in pressure

(a)

$\begin{array}{rcl}\Delta p& =& -\beta \frac{\Delta V}{V_0}\\ & =& -\left(15\times {10}^{9}Pa\right)\left(-0.0010\right)\\ & =& 1.5\times {10}^{7}Pa\\ & =& 150atm\end{array}$

Step 4: Depth increases due to pressure variation

$$\begin{gathered} \because \Delta p=1.5\times {10}^{7}Pa \\ 1atm=1.01\times {10}^5\text{ Pa} \end{gathered}$$

So depth increases is 1.5 Km