Steel wire of lenght ‘L’ at 40 °C is suspended from the ceiling and then a mass ‘m’ is hung from its free end. The wire is cooled down from 40 °C to 30 °C to regain its original length ‘L’. The coefficient of linear thermal expansion of the steel is 10 -5/°C, Young’s modulus of steel is 10 11 N/m 2 and radius of the wire is 1 mm. Assume that L diameter of the wire. Then the value of ‘m’ in kg is nearly

To solve the problem of determining the mass 'm' that can be hung from a steel wire, which cools from 40 °C to 30 °C and regains its original length, we need to apply the concepts of thermal expansion and material properties like Young's modulus. Let's break this down step by step.

Understanding Thermal Expansion

When a material, such as steel, is cooled, it contracts. The amount of contraction can be calculated using the formula for linear thermal expansion:

• ΔL = α * L0 * ΔT

Where:

• ΔL is the change in length.
• α is the coefficient of linear thermal expansion (in this case, 10^-5/°C).
• L0 is the original length of the wire (L).
• ΔT is the change in temperature.

Here, ΔT = 30 °C - 40 °C = -10 °C. Plugging in these values:

• ΔL = 10^-5 * L * (-10) = -10^-4 * L

This means the wire contracts by 10^-4 * L when cooled from 40 °C to 30 °C.

Equilibrium of Forces

Next, we need to consider the forces involved when the mass 'm' is hung from the wire. The tension in the wire must balance the weight of the mass and the restoring force due to the contraction of the wire:

• T = m * g

Where g is the acceleration due to gravity (approximately 9.81 m/s²). The tension can also be expressed in terms of Young's modulus:

• T = (Y * A * ΔL) / L0

Here:

• Y is Young's modulus (10¹¹ N/m² for steel).
• A is the cross-sectional area of the wire, which can be calculated using the radius (1 mm = 0.001 m):

Cross-sectional area A = π * r² = π * (0.001 m)² ≈ 3.14 * 10^-6 m².

Setting Up the Equation

Now we can equate the two expressions for tension:

• m * g = (Y * A * ΔL) / L0

Substituting the values we have:

• m * 9.81 = (10¹¹ N/m² * 3.14 * 10^-6 m² * (-10^-4 * L)) / L

Notice that L cancels out from both sides of the equation:

• m * 9.81 = -10¹¹ * 3.14 * 10^-6 * 10^-4

Now, simplifying this gives:

• m * 9.81 = -3.14 * 10⁷

Finally, solving for 'm':

• m = -3.14 * 10⁷ / 9.81

Calculating this yields:

• m ≈ 3.20 * 10⁶ kg

However, since mass cannot be negative, we interpret this in the context of the required tension to restore the original length against the contraction force. Therefore, the approximate mass that can be hung, considering the contraction and the physical properties of steel, is about:

This value represents the mass that balances the tension created by the wire's contraction when cooled, maintaining the wire's original length.