A metal shaft has a diameter of 5.015 cm at 50^ C . A bearing has an inner diameter of 5.000 cm at the same temperature. The bearing is heated so that the shaft exactly slides into it. If the bearing must be heated to 250^ C for this exact fit, what is the coefficient of linear expansion of the bearing material? (Assume the shaft remains at 50^ C during the process)

Step-by-step solution

The initial inner diameter of the bearing is D₀ = 5.000 cm at T₁ = 50^ C . For the shaft to exactly slide in, the final inner diameter of the bearing must be equal to the diameter of the shaft, so D = 5.015 cm . The required expansion in the diameter is: D = D - D₀ = 5.015 - 5.000 = 0.015 cm The change in temperature of the bearing is: T = T₂ - T₁ = 250^ C - 50^ C = 200^ C Using the formula for linear expansion, D = D₀ T , we can solve for the coefficient of linear expansion : = D D₀ T = 0.015 5.000 200 = 0.015 100