Question

If the temperature of a wire of length $2$ m and area of cross-section $1$ cm$^{2}$ is increased from $0^{\circ}$ C to $80^{\circ}$ C, then force required so that the wire is not allowed to increase in length is: (Given $Y=10^{10}$ N/m$^{2}$, $a=10^{-6}$ $^{\circ}$C) A $160$ N B $80$ N C $400$ N D $120$ N

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Answer

:

The correct answer is B: $80 , \text{N}$.

To determine the force required to prevent the wire from extending due to thermal expansion, we use the formula:

$$F = Y \cdot A \cdot a \cdot \Delta \theta$$

Here,

  • $Y$ represents the Young's modulus of the material.
  • $A$ is the cross-sectional area of the wire.
  • $a$ symbolizes the coefficient of linear expansion.
  • $\Delta \theta$ denotes the change in temperature.

Given:

  • $\Delta \theta = 80^{\circ} \text{C} - 0^{\circ} \text{C} = 80^{\circ} \text{C}$
  • $A = 1 , \text{cm}^2 = 0.0001 , \text{m}^2$ (since $1 , \text{cm}^2 = 0.0001 , \text{m}^2$)
  • $a = 10^{-6} , ^{\circ}\text{C}^{-1}$
  • $Y = 10^{10} , \text{N/m}^2$

Substitute these values into the formula:

$$ F = 10^{10} , \text{N/m}^2 \times 0.0001 , \text{m}^2 \times 10^{-6} , ^{\circ}\text{C}^{-1} \times 80^{\circ} \text{C} $$

Calculating step-by-step:

$$ F = 10^{10} \times 0.0001 \times 10^{-6} \times 80 $$

Simplify this expression:

$$ F = 80 , \text{N} $$

Thus, the force required to prevent the wire from lengthening due to an increase in temperature of $80^\circ\text{C}$ is $80 , \text{N}$.


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