A brass rod is 185 cm long and 1.60 cm in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from 120.0°C to 10.0°C?

Verified step by step guidance

1. First, understand that the problem involves thermal contraction. When the temperature of a material decreases, it contracts. The force needed to prevent this contraction can be calculated using the concept of thermal stress.
2. Calculate the change in temperature (ΔT) using the formula: ΔT = T_final - T_initial. Here, T_final is 10.0°C and T_initial is 120.0°C.
3. Determine the change in length (ΔL) due to thermal contraction using the formula: ΔL = L_initial * α * ΔT, where L_initial is the initial length of the rod, α is the coefficient of linear expansion for brass, and ΔT is the change in temperature.
4. Calculate the stress (σ) required to prevent contraction using the formula: σ = E * (ΔL / L_initial), where E is the Young's modulus for brass.
5. Finally, calculate the force (F) needed using the formula: F = σ * A, where A is the cross-sectional area of the rod. The cross-sectional area can be calculated using the formula: A = π * (d/2)^2, where d is the diameter of the rod.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For solids like a brass rod, this is typically linear expansion, where the length changes proportionally with temperature. The change in length can be calculated using the formula ΔL = αL₀ΔT, where α is the coefficient of linear expansion, L₀ is the original length, and ΔT is the temperature change.

Young's Modulus

Young's modulus is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material. For a rod under tension or compression, the force required to prevent deformation can be calculated using the formula F = (Y * A * ΔL) / L₀, where Y is Young's modulus, A is the cross-sectional area, and ΔL is the change in length.

Stress and Strain

Stress is the force applied per unit area within materials, while strain is the deformation or displacement it causes. In the context of preventing contraction, stress is applied to counteract the thermal contraction. The relationship between stress and strain is linear for small deformations, described by Hooke's Law, which is crucial for calculating the force needed to maintain the rod's original length.