Question

The higher and lower fixed points on a thermometer are separated by $150{\text{mm}}$. Find the temperature reading when the length of the mercury thread is above the lower temperature by $30{\text{mm}}$ .
A) $20^\circ {\text{C}}$
B) $30^\circ {\text{C}}$
C) $10^\circ {\text{C}}$
D) $0^\circ {\text{C}}$

Answer 1

Hint:The upper fixed point corresponds to the full-scale reading of the thermometer and the lower fixed point corresponds to the zero reading of the thermometer. The least count of the thermometer is then found out to determine the temperature reading corresponding to the given length of the mercury thread.

Formula used:
-The temperature in a thermometer is given by, $C = 100 \times \dfrac{{\left( {P - LFP} \right)}}{{\left( {UFP - LFP} \right)}}$ where $P$ is the fixed point corresponding to the mercury thread, ${\text{LFP}}$ is the lower fixed point and ${\text{UFP}}$ is the upper fixed point.

Complete step by step answer.
Step 1: List the parameters given in the question.
We assume the lower fixed point to be at $LFP = 0{\text{mm}}$. Then the upper fixed point will be at $UFP = 150{\text{mm}}$.
The fixed point corresponding to the mercury thread is taken as $P = 30{\text{mm}}$ .
Step 2: Obtain the range of the thermometer so that we can find the temperature reading
In the given thermometer, the upper fixed point is at $UFP = 150{\text{mm}}$. We consider the temperature range of the thermometer to be $\left( {0 - 10{\text{0}}} \right)^\circ {\text{C}}$. So $UFP = 150{\text{mm}}$ corresponds to $100^\circ {\text{C}}$ .
Then a height of $1{\text{mm}}$ will correspond to $\dfrac{{100^\circ {\text{C}}}}{{UFP}} = \dfrac{{100}}{{150}}$. This is the least count of the thermometer.
Then the height of the mercury thread will correspond to $T = \dfrac{{100^\circ {\text{C}}}}{{150}} \times P$ -------- (1)
Substituting for $P = 30{\text{mm}}$ in equation (1) we get, $T = \dfrac{{100^\circ {\text{C}}}}{{150}} \times 30 = 20^\circ {\text{C}}$
Thus we obtain the temperature reading corresponding to the length of the mercury thread as $T = 20^\circ {\text{C}}$ .

So the correct option is A.

Note: Alternate method
Here we assume the lower fixed point to be at $LFP = 0{\text{mm}}$ and the upper fixed point is thus at $UFP = 150{\text{mm}}$so that the distance between the upper fixed point and the lower fixed point remains to be $150{\text{mm}}$ .
The fixed point corresponding to the mercury thread is taken as $P = 30{\text{mm}}$ .
Let $T$ be the temperature reading corresponding to the length of the mercury thread.
The temperature in a thermometer is given by, $T = 100 \times \dfrac{{\left( {P - LFP} \right)}}{{\left( {UFP - LFP} \right)}}$ --------- (1)
Substituting for $P = 30{\text{mm}}$, $LFP = 0{\text{mm}}$ and $UFP = 150{\text{mm}}$ in equation (1) we get, $T = 100 \times \dfrac{{\left( {30 - 0} \right)}}{{\left( {150 - 0} \right)}} = 20^\circ {\text{C}}$
Thus the temperature reading will be $T = 20^\circ {\text{C}}$ .
So the correct option is A.