Final answer:
To fit the band over the barrel, it needs to be heated to 10 °C. The tension in the band when it cools to 20 °C is 2.4 MPa.
Step-by-step explanation:
To determine the temperature to which the band needs to be heated so that it will fit over the barrel, we can use the equation for thermal expansion:
ΔL = αL0ΔT
where ΔL is the change in length, α is the coefficient of linear expansion, L0 is the original length, and ΔT is the change in temperature. In this case, we need to find the change in temperature, so we rearrange the equation:
ΔT = (ΔL) / (αL0)
Substituting the given values: ΔL = (134.110 cm - 134.122 cm) = -0.012 cm
α (iron) = 0.000012 / °C (Given)
L0 = 134.122 cm
Plugging in these values, we can calculate the change in temperature:
ΔT = (-0.012 cm) / (0.000012 / °C * 134.122 cm)
ΔT = -10 °C
Therefore, the band needs to be heated to a temperature of 10 °C so that it will fit over the barrel.
To find the tension in the band when it cools to 20 °C, we can use the equation for thermal stress:
σ = βEΔT
where σ is the stress, β is the coefficient of linear expansion, E is the modulus of elasticity, and ΔT is the change in temperature. Since the band is iron, we can use the following values:
β (iron) = 0.000012 / °C (Given)
E (iron) = 200 GPa (Given)
ΔT = (20 °C - 10 °C) = 10 °C
Plugging in these values, we can calculate the stress:
σ = (0.000012 / °C * 200 GPa * 10 °C)
σ = 2.4 MPa
Therefore, the tension in the band when it cools to 20 °C is 2.4 MPa.