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Found in: Page 439

### Physics For Scientists & Engineers

Book edition 9th Edition
Author(s) Raymond A. Serway, John W. Jewett
Pages 1624 pages
ISBN 9781133947271

# Review. A solid sphere of brass (bulk modulus of ${\mathbf{14}}{\mathbf{.0}}{\mathbf{×}}{{\mathbf{10}}}^{{\mathbf{10}}}{\mathbf{\text{\hspace{0.17em}}}}\mathbf{\text{N}}}{{\mathbf{\text{m}}}^{\mathbf{\text{2}}}}$) with a diameter of ${\mathbf{3}}{\mathbf{.00}}{\mathbf{\text{\hspace{0.17em}m}}}$ is thrown into the ocean. By how much does the diameter of the sphere decrease as it sinks to a depth of ${\mathbf{1}}{\mathbf{\text{\hspace{0.17em}km}}}$?

The decrease in diameter is $\Delta D=2.66×{10}^{-3}\text{\hspace{0.17em}m}$.

See the step by step solution

## Given Data:

The magnetic field, $B=14.0×10{}^{10}\text{\hspace{0.17em}}\text{N}}{{\text{m}}^{\text{2}}}$

The depth of ocean, $h=1\text{km}=1000\text{m}$

The diameter of sphere, $D=3.0\text{\hspace{0.17em} m}$

$\begin{array}{rcl}r& =& \frac{D}{2}\\ & =& \frac{3.0\text{\hspace{0.17em}}\text{m}}{2}\\ & =& 1.5\text{\hspace{0.17em}m}\end{array}$

## A concept:

The pressure $P$ in a fluid is the force per unit area exerted by the fluid on a surface:

$P=\rho gh$

In the SI system, pressure has units of Newton’s per square meter $N}{{m}^{2}}$, and $1\text{N}}{{\text{m}}^{\text{2}}}=1\text{Pascal}\left(\text{Pa}\right)$.

Bulk modulus is given by,

role="math" localid="1663696968659"

## Find the decrease in diameter:

Applying the formula of Bulk modulus here, you get

While pressure is,

$P=\rho gh$

Combining both, you get

$\begin{array}{rcl}{\left(\Delta r\right)}^{3}& =& \frac{\rho gh×{r}^{3}}{B}\\ & =& \frac{1000×10×1000×{\left(1.5\right)}^{3}}{14.0×{10}^{10}}\\ & =& 2.4×{10}^{-4}\end{array}$

$\begin{array}{rcl}\Delta r& =& \sqrt[3]{2.4×{10}^{-4}}\\ & =& 0.062\text{m}\end{array}$

Also diameter, $D=2r$

Therefore decrease in diameter is,

$\begin{array}{rcl}\Delta D& =& 2\Delta r\\ & =& 2×0.062\text{m}\\ & =& 0.124\text{m}\end{array}$

Hence, the decrease in diameter is $\Delta D=0.124\text{m}$.