Question

A piece of brass (alloy of copper and zinc) weighs 12.9g in air. When completely immersed in water it weighs 11.3g. What is the mass of copper contained in the alloy? Specific gravities of copper and zinc are 8.9 and 7.1 respectively:
A. 1.67 g
B. 7.61 g
C. 6.11 g
D. 7.16 g

Answer 1

Hint: When we place something in a liquid, either it sinks or it floats. Every fluid exerts an upward force on objects lying inside it. This upward force is known as buoyant force or the force of buoyancy. The magnitude of this force depends only upon the density of liquid. If this force is more than the weight of the object, it will float and if the force is less than the weight of the object, it will sink.

Formula used:
Weight of liquid (of density ($\sigma$)) displaced = $\sigma Vg$, $F_{Buoyancy} = \sigma Vg$.

Complete answer:
Let the mass of copper in the alloy be $m_c$and that in zinc be $m_z$, then from the question, $m_c+m_z = 12.9 g$
Now, by Archimedes principle, Weight of liquid (of density ($\sigma$)) displaced by an object when immersed fully in the liquid = $\sigma Vg$
Since $F_{Buoyancy}$is equal to the difference in weight of body in air and liquid:
Hence difference in weight = $13.3 - 12.9 = 1.6g = \sigma Vg$
Or $V = \dfrac{1.6g}{1\times g} = 1.6 cm^3$ [as $\sigma = 1g/cm^3$,for water]
Now, the total volume of both the copper and zinc must be equal to total volume (V) of allow:
Thus $V = V_c +V_z$
Now using $\rho=\dfrac mV$,
$V = \dfrac{m_c}{\rho_c} + \dfrac{m_z}{\rho_z} =1.6$
Now, relative density = $\dfrac{density\ of\ substance}{density\ of\ water}$
Hence $\rho_c = 8.9 \times 1 g/cm^3\ = 8.9 g/cm^3 and\ \rho_z = 7.1\times 1g/cm^3 = 7.1 g/cm^3$ [as $density\ of\ water\ = 1g/cm^3$]
Hence putting the values:
$\dfrac{m_c}{\rho_c} + \dfrac{m_z}{\rho_z}=\dfrac{m_c}{8.9} + \dfrac{m_z}{7.1} =1.6$
Also, $m_c+m_z = 12.9 g$
Or $m_z = 12.9 - m_c$
Putting this equation in $\dfrac{m_c}{8.9} + \dfrac{m_z}{7.1} =1.6$
$ \Rightarrow \dfrac{m_c}{8.9} + \dfrac{12.9 - m_c}{7.1} =1.6$
Or $m_c\left(\dfrac{1}{8.9} - \dfrac{1}{7.1}\right) + \dfrac{12.9}{7.1} =1.6$
$ \Rightarrow m_c\left(\dfrac{7.1-8.9}{8.9\times 7.1}\right)=1.6 - \dfrac{12.9}{7.1}$
$\Rightarrow m_c\left(\dfrac{7.1-8.9}{8.9\times 7.1}\right)= \dfrac{1.6\times 7.1 -12.9}{7.1}$
$\Rightarrow m_c=8.9 \left( \dfrac{11.36-12.9}{-1.8}\right)$
$\Rightarrow m_c=8.9 \left( \dfrac{-1.54}{-1.8}\right)= 7.61g$
Hence, option B is correct.

Note: Archimedes principle states that if a body is immersed fully or partially inside a liquid, then it experiences an upward force, which decreases the net weight of the body inside the fluid or liquid. This force is called Buoyant force and it is equal to the weight of the volume liquid displaced by the body. In fully immersed condition, it is equal to the weight of liquid or volume that of the body.