An iron rod is1.58m long at 0°c.what must be the length of a brass rod at 0°c if the difference between the length of the two rods is to remain the same as all temperatures.linear expansivity of iron=1.2×10*-5k-1, linear expansivity of brass=1.9×10*-5k-1
To find the length of the brass rod at 0°C, we can use the concept of linear expansivity. The linear expansivity of a material represents how much the length of the material changes for a one-degree increase in temperature.
Given:
- Length of the iron rod at 0°C: 1.58 m
- Linear expansivity of iron (α_iron): 1.2 × 10^-5 K^-1
- Linear expansivity of brass (α_brass): 1.9 × 10^-5 K^-1
Let's assume the length of the brass rod at 0°C is L_brass (unknown).
We know that the change in length (ΔL) for both materials can be calculated using the formula:
ΔL = α × L_0 × ΔT
Where:
- ΔL: change in length
- α: linear expansivity
- L_0: original length
- ΔT: change in temperature
Since the difference between the lengths of the two rods is to remain the same at all temperatures, we can set up the following equation:
ΔL_iron = ΔL_brass
We can substitute the given values and formulas into the equation:
α_iron × L_iron × ΔT = α_brass × L_brass × ΔT
Since ΔT = 0 (both rods are at 0°C), we can remove the ΔT term from the equation:
α_iron × L_iron = α_brass × L_brass
Now, we can solve for L_brass:
L_brass = (α_iron / α_brass) × L_iron
Plug in the given values:
L_brass = (1.2 × 10^-5 K^-1 / 1.9 × 10^-5 K^-1) × 1.58 m
Calculating:
L_brass = (0.631578947368421 / 1.9) × 1.58 m
L_brass ≈ 0.33 m (rounded to two decimal places)
Therefore, the length of the brass rod at 0°C should be approximately 0.33 meters to maintain the same difference in length as the iron rod.