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Two uniform brass rods A and B of length l and 2l and radii 2r respectively are heated to the same temperature. The ratio of the increase in the volume of A to that of B is
Options
A.1:1
B.1:2
C.2:1
D.4:1

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Last updated date: 13th Jul 2024
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Answer
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Hint: Brass is a copper-zinc alloy with variable amounts of copper and zinc to produce different mechanical and electrical properties. It's a substitutional alloy, which means the atoms from the two constituents will swap places within the same crystal structure.

Complete answer:
The atoms in a solid vibrate faster about their fixed points as it is heated. As a result, when solids are heated, the relative increase in size is negligible. When the sun heats metal train rails, they stretch through the cracks and do not buckle.
The tendency of an object to change its dimension in length, field, or volume as a result of heat is known as thermal expansion. The kinetic energy of a material increases as it is heated. There are three types of thermal expansion, depending on the type of expansion:
Linear expansion
Area expansion
Volume expansion
The increase in volume caused by temperature is known as volume expansion. The formula for volume expansion is as follows:
\[\dfrac{{\Delta V}}{{{V_o}}} = {\alpha _V}\Delta T\]
\[{V_o}\] = original volume,
V = expanded volume,
αv = volume expansion coefficient,
ΔT = temperature difference,
ΔV = change in volume after expansion
Let the original temperature be 0⁰ C
\[A = l \times \pi {(2r)^2} = {V_1}\]
\[V_1^\prime = {V_1}(1 + \gamma \Delta T)\]
\[\dfrac{{\left( {V' - {V_1}} \right)}}{{{V_1}}} = \gamma \Delta T\]
\[ \Rightarrow {V_1}' - {V_1} \propto {V_1}\]
\[\dfrac{{\left( {{V_2}' - {V_2}} \right)}}{{{V_2}}} = \gamma \Delta T\]
\[ \Rightarrow {V_2}' - {V_2} \propto {V_2}\]
\[\therefore \quad \dfrac{{\Delta {V_1}}}{{\Delta {V_2}}} = \dfrac{{l{{(2r)}^2}}}{{2l{r^2}}} = \dfrac{2}{1}\]
Ratio = 2:1

Hence, option C is correct.

Note:
Since heat allows molecules to travel quickly (heat energy is converted to kinetic energy), the volume of a gas expands more than that of a solid or liquid.
Gases trapped in a fixed volume, on the other hand, cannot multiply, so temperature increases result in pressure increases.