10.9 A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit?

Height of the spirit column, $h_1$ = 12.5 cm = 0.125 m

Height of the water column, $h_2$ = 10 cm = 0.1m

$P_0$ = atmospheric pressure

${\rho }_1$ = density of spirit, ${\rho }_2$ = density of water

Pressure at point B = $P_0$ + $h_1\mathrm{}{\rho }_1\mathrm{}\mathrm{g}$

Pressure at point D = $P_0$ + $h_2\mathrm{}{\rho }_2\mathrm{}\mathrm{g}$

But pressure at point B and D are same. Hence

$P_0$ + $h_1\mathrm{}{\rho }_1\mathrm{}\mathrm{g}$ = $P_0$ + $h_2\mathrm{}{\rho }_2\mathrm{}\mathrm{g}$ or $h_1\mathrm{}{\rho }_1$ = $h_2\mathrm{}{\rho }_2$

$\frac{\mathrm{}{\rho }_1}{{\rho }_2}$ = $\frac{\mathrm{}{h}_2}{h_1}$ = $\frac{10}{12.5}$ = 0.8

So the specific gravity of spirit is 0.8

<p>Height of the spirit column,&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;= 12.5 cm = 0.125 m</p><p>Height of the water column,&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>&nbsp;= 10 cm = 0.1m</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;= atmospheric pressure</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;= density of spirit,&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span>&nbsp;= density of water</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728392845phpbOBP2D_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728392845phpbOBP2D.jpeg" alt="" width="172" height="138"></picture></div></div><p>Pressure at point B =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;+&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi></math></span></p><p>Pressure at point D =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;+&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi></math></span></p><p>But pressure at point B and D are same. Hence</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;+&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>P</mi></mrow></mrow><mrow><mrow><mn>0</mn></mrow></mrow></msub></math></span>&nbsp;+&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi></math></span>&nbsp;or&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></math></span></p><p><span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>ρ</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi mathvariant="normal"></mi><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msub></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>12.5</mn></mrow></mrow></mfrac></math></span>&nbsp;= 0.8</p><p>So the specific gravity of spirit is 0.8</p>

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