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Question: If a cylinder of diameter \[1.0cm\]at \[{30^o}C\]is to be slid into a hole of diameter \[0.9997cm\]i...

If a cylinder of diameter 1.0cm1.0cmat 30oC{30^o}Cis to be slid into a hole of diameter 0.9997cm0.9997cmin a steel plate at the same temperature, the minimum required rise in the temperature of the plate is: (coefficient of linear expansion of steel=12×10−6/oC = 12 \times {10^{ - 6}}{/^o}C
(A) 25oC{25^o}C
(B) 35oC{35^o}C
(C ) 45oC{45^o}C
(D) 55oC{55^o}C

Explanation

Solution

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 = 1cm
Thus, the increase in diameter of the hole is=1cm−= 1cm -Initial diameter of the hole
So, Change in Diameter is =1cm−0.9997cm=0.003cm = 1cm - 0.9997cm = 0.003cm
Now, let us calculate the required rise in temperature.
We know, the formula of thermal expansion:

Δd=doαΔt\Delta d = {d_o}\alpha \Delta t

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

Putting the values as given in the question:

0.003=0.9997×12×10−6×Δt0.003 = 0.9997 \times 12 \times {10^{ - 6}} \times \Delta t

Rearranging the equation, we obtain:

Δt=0.0030.9997×12×10−6C\Delta t = \dfrac{{0.003}}{{0.9997 \times 12 \times {{10}^{ - 6}}}}C

Hence, we obtain:

Δt=25oC\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.