a thin rod of A thin rod of negligible mass and are of cross-section 4x10^–6 m2, suspended vertically from one end,
has a length of 0.5 m at 100°C. The rod is cooled to 0°C, but prevented from contracting by attaching a
mass at the lower end. (Given Y = 10^11 Nm–2 the co-efficient of linear expansion α = 10^–5 K–1 and the
value of g is 10 ms–2). Then mass attached and energy stored in the rod are

To solve this problem, we need to analyze the situation involving thermal expansion and the forces acting on the rod when it is cooled. The rod is initially at a temperature of 100°C and has a length of 0.5 m. When it is cooled to 0°C, it would naturally contract due to thermal contraction. However, since it is prevented from doing so by a mass attached at the lower end, we can calculate the mass required and the energy stored in the rod due to this constraint.

Understanding Thermal Expansion

When materials are heated, they expand, and when they are cooled, they contract. The amount of contraction can be calculated using the formula:

ΔL = L₀ * α * ΔT

Where:

• ΔL = change in length
• L₀ = original length (0.5 m)
• α = coefficient of linear expansion (10^-5 K^-1)
• ΔT = change in temperature (100°C to 0°C, so ΔT = -100°C)

Calculating the Change in Length

Substituting the values into the formula:

ΔL = 0.5 m * (10^-5 K^-1) * (-100 K) = -0.0005 m

This means the rod would contract by 0.0005 m (or 0.5 mm) if it were free to do so. However, since it is constrained by the mass, we need to find the force exerted by this mass to keep the rod from contracting.

Force and Mass Calculation

The force exerted by the mass (m) due to gravity can be expressed as:

F = m * g

Where:

• g = acceleration due to gravity (10 m/s²)

To find the stress in the rod due to the applied force, we can use the formula:

Stress = F / A

Where A is the cross-sectional area of the rod (4 x 10^-6 m²). The stress can also be related to strain using Young's modulus (Y):

Stress = Y * Strain

Strain is defined as the change in length divided by the original length:

Strain = ΔL / L₀ = -0.0005 m / 0.5 m = -0.001

Calculating the Stress

Now we can calculate the stress:

Stress = Y * Strain = (10¹¹ N/m²) * (-0.001) = -10⁸ N/m²

Finding the Mass

Now we can set the two expressions for stress equal to each other:

m * g / A = -10⁸ N/m²

Substituting the values:

m * 10 m/s² / (4 x 10^-6 m²) = 10⁸ N/m²

Solving for m gives:

m = (10⁸ N/m²) * (4 x 10^-6 m²) / (10 m/s²) = 4 kg

Energy Stored in the Rod

The energy stored in the rod due to the stress can be calculated using the formula:

U = (1/2) * Stress * Strain * Volume

First, we need to calculate the volume of the rod:

Volume = A * L₀ = (4 x 10^-6 m²) * (0.5 m) = 2 x 10^-6 m³

Now substituting the values into the energy formula:

U = (1/2) * (10⁸ N/m²) * (-0.001) * (2 x 10^-6 m³)

Calculating this gives:

U = (1/2) * (10⁸) * (-0.001) * (2 x 10^-6) = -0.1 J

Final Results

To summarize:

• The mass attached to the rod is 4 kg.
• The energy stored in the rod due to the stress is -0.1 J.

This negative sign indicates that the energy is stored in a compressive state, which is typical when a material is under tension or compression beyond its natural length. Understanding these principles helps in various engineering applications, especially in materials science and structural engineering.