A steel rod is 3cm in diameter at 25C. A brass ring has an interior diameter hole of 2.994cm at 25C. At what common temp will the ring just slide onto the rod?
alpha(steel)- 11x10^-6
alpha(brass)= 19x10^-6
To find the common temperature at which the brass ring will just slide onto the steel rod, we need to consider the thermal expansion of both materials.
The change in diameter of the steel rod (ΔD_steel) can be calculated using the formula:
ΔD_steel = α_steel * D_steel * ΔT
Where:
α_steel is the coefficient of linear expansion for steel (given as 11x10^-6)
D_steel is the initial diameter of the steel rod (3cm)
ΔT is the change in temperature.
Similarly, the change in diameter of the brass ring (ΔD_brass) can be calculated using the formula:
ΔD_brass = α_brass * D_brass * ΔT
Where:
α_brass is the coefficient of linear expansion for brass (given as 19x10^-6)
D_brass is the initial interior diameter of the brass ring (2.994cm)
Since we want the ring to just slide onto the rod, the change in diameter of the ring should be equal to the change in diameter of the rod. Therefore, we can set up the following equation:
ΔD_steel = ΔD_brass
Solving for ΔT:
α_steel * D_steel * ΔT = α_brass * D_brass * ΔT
Now, we can cancel out ΔT from both sides of the equation:
α_steel * D_steel = α_brass * D_brass
Substituting the given values:
11x10^-6 * 3 = 19x10^-6 * 2.994
Simplifying:
33x10^-6 = 57.186x10^-6
Finally, solving for ΔT:
ΔT = (57.186x10^-6) / (11x10^-6)
ΔT ≈ 5.20°C
Therefore, at a common temperature of approximately 5.20°C, the brass ring will just slide onto the steel rod.