A circular disc made of iron is rotated about its axis at a constant velocity v. Calculate the percentage change in the linear speed of a particle of the rim as the disc is slowly heated from 20 C to 50 C keeping the angular velocity constant. Coefficient of linear expansion of iron = 1.2 * 10^(-5)/C

To tackle the problem of calculating the percentage change in the linear speed of a particle on the rim of a circular iron disc as it is heated, we need to understand how temperature affects the dimensions of the disc and subsequently the linear speed of the particles on its edge. Since the angular velocity remains constant, the change in linear speed will be directly related to the change in the radius of the disc due to thermal expansion.

Understanding Thermal Expansion

When materials are heated, they tend to expand. This phenomenon is quantified by the coefficient of linear expansion, which for iron is given as 1.2 × 10^-5/°C. This coefficient indicates how much a unit length of iron will expand for each degree Celsius increase in temperature.

Calculating the Change in Radius

The formula for linear expansion is:

• ΔL = L₀ × α × ΔT

Where:

• ΔL = change in length (or radius, in this case)
• L₀ = original length (or radius)
• α = coefficient of linear expansion
• ΔT = change in temperature

In our scenario:

• Let L₀ = r (original radius of the disc)
• ΔT = 50°C - 20°C = 30°C

Substituting the values into the formula gives:

• Δr = r × (1.2 × 10^-5) × 30

This results in:

• Δr = r × 3.6 × 10^-4

New Radius Calculation

The new radius after heating will be:

• r_new = r + Δr = r + r × 3.6 × 10^-4 = r(1 + 3.6 × 10^-4)

Linear Speed and Angular Velocity

The linear speed (v) of a particle on the rim of the disc is related to the radius and angular velocity (ω) by the formula:

• v = r × ω

Since the angular velocity remains constant, the new linear speed (v_new) after heating can be expressed as:

• v_new = r_new × ω = r(1 + 3.6 × 10^-4) × ω

Percentage Change in Linear Speed

The percentage change in linear speed can be calculated using the formula:

• Percentage Change = ((v_new - v) / v) × 100%

Substituting the expressions for v and v_new gives:

• Percentage Change = (((r(1 + 3.6 × 10^-4) × ω) - (r × ω)) / (r × ω)) × 100%

After simplifying, we find:

• Percentage Change = (3.6 × 10^-4) × 100% = 0.036%

Final Result

Thus, the percentage change in the linear speed of a particle on the rim of the disc as it is heated from 20°C to 50°C, while keeping the angular velocity constant, is approximately 0.036%.