Question

If a cylinder of diameter \[1.0cm\]at \[{30^o}C\]is to be slid into a hole of diameter \[0.9997cm\]in a steel plate at the same temperature, the minimum required rise in the temperature of the plate is: (coefficient of linear expansion of steel\[ = 12 \times {10^{ - 6}}{/^o}C\]
(A) \[{25^o}C\]
(B) \[{35^o}C\]
(C ) \[{45^o}C\]
(D) \[{55^o}C\]

Answer 1

Hint:We need to find the rise in temperature that can cause the hole in the steel plate to have the same diameter as that of the cylinder.
From the given diameters we first need to calculate the increase in diameter that is required in the hole. Applying the formula for thermal expansion we can then calculate the rise in temperature from the values of thermal coefficient and the rise in diameter given.

Complete Step-By-Step Solution:
As we can infer from the question, we need to find the increase in diameter of the hole such that the cylinder may fit in.
It is clear that for the cylinder to slide inside the hole of the steel plate, diameter of the cylinder as well as diameter of the hole in the steel plate must be equal.
Therefore, after expansion, the diameter of hole must be \[ = 1cm\]
Thus, the increase in diameter of the hole is\[ = 1cm - \]Initial diameter of the hole
So, Change in Diameter is \[ = 1cm - 0.9997cm = 0.003cm\]
Now, let us calculate the required rise in temperature.
We know, the formula of thermal expansion:

\[\Delta d = {d_o}\alpha \Delta t\]

Where,
\[\Delta d = \]Change in diameter of the hole \[{d_o} = \] Initial diameter of the hole
\[\alpha = \] Coefficient of linear expansion of steel
\[\Delta t = \] Rise in temperature

Putting the values as given in the question:

\[0.003 = 0.9997 \times 12 \times {10^{ - 6}} \times \Delta t\]

Rearranging the equation, we obtain:

\[\Delta t = \dfrac{{0.003}}{{0.9997 \times 12 \times {{10}^{ - 6}}}}C\]

Hence, we obtain:

\[\Delta t = {25^o}C\]

This is the required answer, hence option (A) is correct.

Note:Thermal expansion is a tendency of matter to change its state, area, volume. Thermal expansion occurs when due to application of heat, the molecules present states moving vigorously and taking up more space. This change in dimensions occurs over a narrow interval of temperature that is why it is referred to as linear coefficient of expansion.