Question

The temperature of two bodies measured by a thermometer are ${t_1} = {{2}}{{{0}}^{\circ}}\;{{C \pm 0}}{{.}}{{{5}}^{\circ}}\;{{C}}$ and ${t_2} = 5{{{0}}^{\circ}}\;{{C \pm 0}}{{.}}{{{5}}^{\circ}}\;{{C}}$. Calculate the temperature difference and error there.

Answer 1

While calculating the temperature difference, the change in temperature is found by calculating the difference in two temperatures and adding up the errors in two temperatures. If the temperature of the final state increases compared to the initial state then the change in temperature would be positive. Similarly, the change in temperature would be negative if there were a decrease in temperature from initial state to final state. The absolute error of change in temperature will be the sum of the absolute errors of temperature of two bodies.

Formula used:
The difference in temperature can be calculated by:
$t = {t_2} - {t_1}$
Where, ${t_1}$ and ${t_2}$ are the temperatures of the two bodies.

Complete Step by step solution:
The data that is given in the problem is,
The temperature of two bodies;
${t_1} = {{2}}{{{0}}^{\circ}}\;{{C \pm 0}}{{.}}{{{5}}^{\circ}}\;{{C}}$
${t_2} = 5{{{0}}^{\circ}}\;{{C \pm 0}}{{.}}{{{5}}^{\circ}}\;{{C}}$
Here the temperature had increased. The change in temperature would be positive.
The difference in temperature is given as,
$t = {t_2} - {t_1} \\\ = 5{{{0}}^{\circ}}\;{{C}} - 2{{{0}}^{\circ}}\;{{C}} \\\ {{ = 3}}{{{0}}^{\circ}}\;{{C}} \\\$
Adding the errors in the temperature, we can find the total error that can occur in the temperature change.
The error in temperature difference is given as,
$\Delta t = \Delta {t_2} + \Delta {t_1} \\\ = {0.5^{\circ}}\;{{C + }}{0.5^{\circ}}\;{{C}} \\\ {{ = }}{{{1}}^{\circ}}\;{{C}} \\\$
Therefore, the temperature difference is ${{3}}{{{0}}^{\circ}}\;{{C}} \pm {{{1}}^{\circ}}\;{{C}}$.

Note:
While measuring there can be differences in the actual value and the measured value. This error can be termed as absolute error. The increase or decrease in the value is added to the measured value for accuracy. If the two measurements are given as $P \pm \Delta P$ and $Q \pm \Delta Q$, where $P,Q$ are the measured values and $\Delta P,\Delta Q$ are the absolute errors respectively, then the difference in measurements $R$ is given as $R = P - Q$. The error calculation is also necessary for accuracy. Therefore, the maximum error will be $\Delta R = \Delta P + \Delta Q$. The absolute errors are to be added. The resultant difference in measurement will be $R \pm \Delta R$.