# If the temperature of a wire of length 2 m and area of cross-section 1 cm^2 is increased from 0 degrees C to 80 degrees 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 degrees C) A 160 N B 80 N C 400 N D 120 N

## 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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