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Expert-verified Found in: Page 409 ### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718 # An iron casting containing a number of cavities weighs 6000 N in air and 4000 N in water. What is the total volume of all the cavities in the casting? The density of iron (that is, a sample with no cavities) is ${\mathbf{7}}{\mathbf{.}}{\mathbf{87}}{\mathbf{}}{\mathbit{g}}{\mathbf{/}}{\mathbit{c}}{{\mathbit{m}}}^{{\mathbf{3}}}$.

The total cavity volume in the casting is $0.126{\mathrm{m}}^{3}$

See the step by step solution

## Step 1: The given data

• The weight of iron casting in air, W=6000 N
• The weight of iron casting in water, ${W}_{eff}=4000N$
• The density of solid iron, ${\rho }_{iron}=7.87\frac{g}{c{m}^{3}}or7.87×{10}^{3}\frac{kg}{{m}^{3}}$

## Step 2: Understanding the concept of Archimedes Principle

We can use the concept of Archimedes’ principle and expression of density. When an iron casting is fully submerged in water, then a buoyant force from the surrounding fluid acts on it. This force has a magnitude equal to the weight of the water displaced by the iron casting.

Formulae:

Force applied on body (or weight), ${F}_{b}={m}_{f}g$ (i)

Density of a substance, $\mathrm{\rho }=\frac{\mathrm{m}}{\mathrm{V}}$ (ii)

## Step 3: Calculation of total volume of cavity

The volume of cavity is the difference between volume of casting and volume of iron. Hence,

${V}_{cavity}={V}_{casting}={V}_{iron}$ (iii)

According to equation (ii), we get

The volume of iron as:

${V}_{iron}=\frac{{m}_{iron}}{{\rho }_{iron}}\phantom{\rule{0ex}{0ex}}=\frac{g{m}_{iron}}{g{\rho }_{iron}}\phantom{\rule{0ex}{0ex}}=\frac{W}{g{\rho }_{iron}}$

When the iron casting is submerged in water, the forces acting on it are ${F}_{g}-{F}_{b}$. Hence,the effective weight in water is given as:

${W}_{eff}=W-g{\rho }_{w}{V}_{casting}\phantom{\rule{0ex}{0ex}}{V}_{casting}=\frac{W-{W}_{eff}}{g{\rho }_{w}}$

Equation (iii) becomes and using the given values, we get

Hence, the volume of cavity is $0.126{m}^{3}$ ### Want to see more solutions like these? 