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| <h3 id="back-link"><a href="../../#4.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#4.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
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| <p> | | <p> |
| $4.2.7$ | | $4.2.7$ |
| A body in water weighs three times less than in air. What is the density of the body? | | A body in water weighs three times less than in air. What is the density of the body? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let's consider the following figure | | Let's consider the following figure |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" alt="4.2.7" width="400" /> | | loading="lazy" alt="4.2.7" width="400" /> |
| <figcaption> | | <figcaption> |
| Force analysis | | Force analysis |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| These normal forces $N_i$ is what the equipment measures as weight in each case. | | These normal forces $N_i$ is what the equipment measures as weight in each case. |
| From Newton Second Law for air | | From Newton Second Law for air |
| $$N_a = mg$$ | | $$N_a = mg$$ |
| Since $m=\rho~V$, | | Since $m=\rho~V$, |
| $$N_a = \rho~V~g \;(1)$$ | | $$N_a = \rho~V~g \;(1)$$ |
| Applying Newton Second Law for water | | Applying Newton Second Law for water |
| $$mg = N_w + F_A$$ | | $$mg = N_w + F_A$$ |
| $$\rho~V~g = N_w + \rho_w~V~g$$ | | $$\rho~V~g = N_w + \rho_w~V~g$$ |
| $$N_w = (\rho-\rho_w)~V~g \;(2)$$ | | $$N_w = (\rho-\rho_w)~V~g \;(2)$$ |
| Dividing $(1)$ by $(2)$, taking account that $\frac{N_a}{N_w} = 3$ | | Dividing $(1)$ by $(2)$, taking account that $\frac{N_a}{N_w} = 3$ |
| $$\rho = \frac{3}{2}\rho_w$$ | | $$\rho = \frac{3}{2}\rho_w$$ |
| as $\rho_w = 1$ g/cm$^3$ | | as $\rho_w = 1$ g/cm$^3$ |
| $$\boxed{\rho = 1.5~\rm{g/cm^3}}$$ | | $$\boxed{\rho = 1.5~\rm{g/cm^3}}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
| </p> | | </p> |
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