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A piece of brass (alloy of copper and zinc) weights \[12.9\,g\] in air. When completely immersed in water it weighs \[11.3\,g\]. Specific gravity of copper and zinc are \[8.9\] and \[7.1\] respectively. The mass of copper contained in the alloy is:

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: To solve the problem think about the Archimedes principle, which says that whenever we immerse a body in any liquid it experiences a force due to the liquid which acts in the upward direction. It is the upthrust or buoyant force which equates the mass of water a body has displaced.

Complete step by step answer:
In the above question a piece of brass is given. We know that brass is an alloy which means it is made up of more than one element. Brass is made up of copper and zinc. As we know, specific gravity is the ratio of density of substance to density of water.
Density of water is \[1\,gc{c^{ - 1}}\].
Density of copper is \[8.9\,gc{c^{ - 1}}\].
Density of zinc is \[7.1\,gc{c^{ - 1}}\].
Weight of a brass piece in air is \[12.9\,g\].
Weight of a brass piece in water when it is fully immersed is \[11.3\,g\].
Specific gravity of copper is \[8.9\] and Specific gravity of zinc is \[7.1\]

To find the mass of copper present in the brass. As the weight of brass decreases in water, the weight of water displaced by the piece of brass is given as:
weight in air \[ - \]weight in water \[ = \] water displaced
\[12.9 - 11.3 = 1.6\]
Weight of water displaced by a piece of brass is \[1.6\,g\]. Let us now assume that the mass of copper contained in the brass piece be \[{m_c}\]. Volume of alloy is the sum of volume of copper and volume of zinc.
\[1.6 = \dfrac{{{m_c}}}{{8.9}} + \dfrac{{12.9 - {m_c}}}{{7.1}}\]
Solving the equation for mass of copper
\[\therefore {m_c} = 7.61\,g\]

Thus, the mass of copper contained in alloy is \[7.61\,g\].

Note: If the body is not fully immersed, that is a part of the body is immersed in liquid then the body only displaces the amount of liquid which is equal to the volume of body part which is immersed in the liquid.
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