KVPY Sample Paper KVPY Stream-SX Model Paper-4

A glass sinker has a mass \[M\]in air. When weighed in a liquid at temperature \[{{t}_{1}},\] the apparent mass is \[{{M}_{1}}\] and when weighed in the same liquid at temperature \[{{t}_{2}}\], the apparent mass is \[{{M}_{2}}\]. If the coefficient of cubical expansion of the glass is \[{{\gamma }_{g}},\] then the real coefficient of expansion of the liquid is:

A) \[{{\gamma }_{g}}+\left( \frac{{{M}_{2}}-{{M}_{1}}}{M-{{M}_{2}}} \right).\frac{1}{\left( {{t}_{2}}-{{t}_{1}} \right)}\]

B) \[{{\gamma }_{g}}-\left( \frac{{{M}_{2}}-{{M}_{1}}}{M-{{M}_{2}}} \right).\frac{1}{\left( {{t}_{2}}-{{t}_{1}} \right)}\]

C) \[{{\gamma }_{g}}-\left( \frac{M-{{M}_{2}}}{{{M}_{2}}-{{M}_{1}}} \right).\frac{1}{\left( {{t}_{2}}-{{t}_{1}} \right)}\]

D) \[{{\gamma }_{g}}+\left( \frac{{{M}_{2}}-{{M}_{1}}}{{{M}_{2}}+{{M}_{1}}} \right).\frac{1}{\left( {{t}_{2}}-{{t}_{1}} \right)}\]

Correct Answer: A

Solution :

[a]

\[{{M}_{1}}g=Mg-{{V}_{1}}\rho {{\ell }_{1}}g\]
or \[{{M}_{1}}g=Mg-{{V}_{1}}{{\rho }_{1}}g\]	...(i)
and \[{{M}_{2}}g=Mg-{{V}_{1}}[1+{{\gamma }_{g}}({{t}_{2}}-{{t}_{1}})]\frac{\rho }{[1+\gamma \ell ({{t}_{2}}-{{t}_{1}})]}g\]	?(ii)
After simplifying, we get \[{{\gamma }_{\ell }}={{\gamma }_{g}}+\left( \frac{{{M}_{2}}-{{M}_{1}}}{M-{{M}_{2}}} \right)\frac{1}{\left( {{t}_{2}}-{{t}_{1}} \right)}\]