Question

Two marks on a glass rod 10cm apart are found to increase their distance by 0.08mm when the rod is heated from ${{0}^{\circ }}C$ to ${{100}^{\circ }}C$. A flask made of the same glass as that of a rod measures a volume of 100cc to ${{0}^{\circ }}C$. The volume it measures at ${{100}^{\circ }}C$ in cc is
(A) 100.24
(B) 100.12
(C ) 100.36
(D) 100.48

Answer 1

Hint: A change in the temperature of a body causes changes in its dimensions. The thermal expansion is increased in the dimension of a body by the increase in its temperature. If the substance is in the form of rod, then for small change in temperature the change in length is directly proportional. The coefficient of volume expansion is thus three times the coefficient of coefficient of linear expansion. Then calculate the final volume.

Formula used:
$\vartriangle l=l\alpha \vartriangle T$ …………(1)
where, $\vartriangle l$ is the change in length
l is the length of the rod
$\alpha $ is the coefficient of thermal expansion
$\vartriangle T$ is the change in temperature
$\begin{align}
  & V={{V}_{0}}\left( 1+\gamma \vartriangle T \right) \\
 & \\
\end{align}$ …………….(2)
Where, ${{V}_{0}}$ is the initial volume.
$\gamma $ is the coefficient of expansion.
$\vartriangle T$ is the change in temperature
V is the final volume.

Complete step by step solution:
Given that ,
$\begin{align}
  & \vartriangle l=0.08mm=0.08\times {{10}^{-1}}cm \\
 & l=10cm \\
 & \vartriangle T=100-0={{100}^{\circ }}C \\
\end{align}$
Then by using the formula (1) and rearranging them we will get an equation for $\alpha $.
$\begin{align}
  & \vartriangle l=l\alpha \vartriangle T \\
 & \Rightarrow \alpha =\dfrac{\vartriangle l}{l\vartriangle T} \\
\end{align}$
Substituting the values we get,
$\begin{align}
  & \Rightarrow \alpha =\dfrac{0.08\times {{10}^{-1}}}{10\times \left( 100-0 \right)} \\
 & \Rightarrow \alpha =8\times {{10}^{-6}} \\
\end{align}$
Then the coefficient of volume expansion is thus three times the coefficient of coefficient of linear expansion.
$\begin{align}
  & \therefore \gamma =3\alpha \\
 & \Rightarrow \gamma =3\times 8\times {{10}^{-6}} \\
 & \therefore \gamma =24\times {{10}^{-6}} \\
\end{align}$
Also given that,
Initial volume, ${{V}_{0}}=100cc$
Thus,
$\begin{align}
  & V={{V}_{0}}\left( 1+\gamma \vartriangle T \right) \\
 & \\
\end{align}$
Where, ${{V}_{0}}$ is the initial volume
$\gamma $ is the coefficient of expansion
$\vartriangle T$ is the change in temperature
V is the final volume.
 Substituting the values we get,
$\begin{align}
  & V=100\times \left( 1+24\times {{10}^{-6}}\times \left( 100-0 \right) \right) \\
 & \Rightarrow V=100\times 1.0024 \\
 & \therefore V=100.24cc \\
\end{align}$

So, the correct answer is “Option A”.

Note: A change in the temperature of a body causes changes in its dimensions. The thermal expansion is increased in the dimension of a body by the increase in its temperature. The expansion in length is called linear expansion. The expansion in the area is called area expansion. The expansion in volume is called volume expansion.