Question

Two wires with the same area of cross section and length but young's modulus E and Y respectively are joined end to end. Find equivalent young's modulus.

• A. (E+Y)/2
• B. EY/(E+Y)
• C. no change
• D. E+Y

Answer 1

Answer: (A)

(E+Y)/2

Answer 2

The problem describes two wires joined end to end. This configuration is a series combination. For wires in series, the force (tension) is the same through both wires, and the total extension is the sum of individual extensions.

Let $L$ be the length and $A$ be the cross-sectional area of each wire. Let $E$ and $Y$ be their Young's moduli.

The extension of the first wire is $\Delta L_1 = \frac{FL}{AE}$.

The extension of the second wire is $\Delta L_2 = \frac{FL}{AY}$.

The total extension is $\Delta L_{total} = \Delta L_1 + \Delta L_2 = \frac{FL}{A} \left(\frac{1}{E} + \frac{1}{Y}\right) = \frac{FL}{A} \frac{E+Y}{EY}$.

For the equivalent wire, the total length is $2L$ and the area is $A$. If $E_{eq}$ is the equivalent Young's modulus, then $\Delta L_{total} = \frac{F(2L)}{AE_{eq}}$.

Equating the expressions for $\Delta L_{total}$: $\frac{F(2L)}{AE_{eq}} = \frac{FL}{A} \frac{E+Y}{EY}$.

This simplifies to $\frac{2}{E_{eq}} = \frac{E+Y}{EY}$, which gives $E_{eq} = \frac{2EY}{E+Y}$.

This result is not among the options.

However, if the wires were joined side-by-side (parallel combination), the extension for both wires would be the same, and the total force would be the sum of individual forces.

For parallel combination: $F_1 = E \frac{A \Delta L}{L}$ and $F_2 = Y \frac{A \Delta L}{L}$.

Total force $F_{total} = F_1 + F_2 = (E+Y) \frac{A \Delta L}{L}$.

For the equivalent wire, the total length is $L$ and the total area is $2A$. $F_{total} = E_{eq} \frac{(2A) \Delta L}{L}$.

Equating the forces: $(E+Y) \frac{A \Delta L}{L} = E_{eq} \frac{(2A) \Delta L}{L}$.

This simplifies to $E+Y = 2E_{eq}$, which gives $E_{eq} = \frac{E+Y}{2}$.

This result matches option (a). Given the options and common exam patterns, it is highly probable that the question intends to describe a parallel combination, despite the wording "joined end to end".