Question

A brass wire of length $2\,\text{m}$ and radius $1\,\text{mm}$ at $27^\circ\text{C}$ is held taut between two rigid supports. Initially it was cooled to a temperature of $-43^\circ\text{C}$ creating a tension $T$ in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to $1.4T$ is ___$^\circ$C.

• A. $-71$
• B. $-65$
• C. $-80$
• D. $-86$

Hint:

For wires with fixed ends, thermal stress (and hence tension) is directly proportional to temperature change.

Answer 1

The Correct Option is A

Step 1: Relation between thermal strain and stress.
For a wire fixed between rigid supports, thermal stress is given by:
\[ \sigma = Y \alpha \Delta T \] Hence, tension is directly proportional to temperature change.
Step 2: Comparing two tension conditions.
\[ \dfrac{T_2}{T_1} = \dfrac{\Delta T_2}{\Delta T_1} \] Given $T_2 = 1.4T_1$.
Step 3: Calculating initial temperature change.
\[ \Delta T_1 = 27 - (-43) = 70^\circ\text{C} \] Step 4: Finding new temperature change.
\[ \Delta T_2 = 1.4 \times 70 = 98^\circ\text{C} \] Step 5: Calculating final temperature.
\[ T = 27 - 98 = -71^\circ\text{C} \] Step 6: Final conclusion.
The wire must be cooled to $-71^\circ$C to make the tension $1.4T$.