An iron rod is 1.58 metre long at 0 degree Celsius, what must be the length of a brass rod at 0 degree Celsius if the difference between the lengths of the two rods is to remain the same at all temperatures.
Linear expansivity of iron = 1.2X10^-5 K^-1
Linear expansivity of brass = 1.9X10^-5 K^-5
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To find the length of the brass rod at 0 degrees Celsius, we need to consider the linear expansivity of both metals and their respective temperature changes.
Let's assume the temperature change is ΔT degrees Celsius for both rods.
The final length of the iron rod can be calculated using the formula for linear expansivity:
ΔL_iron = L_iron * α_iron * ΔT
Where:
ΔL_iron = Change in length of the iron rod
L_iron = Initial length of the iron rod (1.58 meters)
α_iron = Linear expansivity of iron (1.2 x 10^-5 K^-1)
ΔT = Change in temperature in Celsius
Similarly, the final length of the brass rod can be calculated using the same formula:
ΔL_brass = L_brass * α_brass * ΔT
Where:
ΔL_brass = Change in length of the brass rod
L_brass = Initial length of the brass rod (unknown)
α_brass = Linear expansivity of brass (1.9 x 10^-5 K^-1)
ΔT = Change in temperature in Celsius
Since we want the difference in lengths to remain the same at all temperatures, ΔL_iron = ΔL_brass. Therefore:
L_iron * α_iron * ΔT = L_brass * α_brass * ΔT
The ΔT term cancels out:
L_iron * α_iron = L_brass * α_brass
Rearranging the formula, we get:
L_brass = (L_iron * α_iron) / α_brass
Now, we can substitute the known values into the equation:
L_brass = (1.58 * 1.2 x 10^-5) / (1.9 x 10^-5)
Calculating the result, we find:
L_brass = 1.008 meters
Therefore, the length of the brass rod at 0 degrees Celsius must be approximately 1.008 meters to maintain the same difference in lengths with the iron rod at all temperatures.
To find the length of the brass rod at 0 degrees Celsius that will maintain the same difference in length with the iron rod at different temperatures, we need to use the concept of linear expansivity.
First, let's find the change in length of the iron rod as the temperature changes. We know that the linear expansivity of iron is 1.2 x 10^-5 K^-1.
The formula for linear expansion is:
ΔL = αLΔT
where ΔL is the change in length, α is the linear expansivity, L is the original length, and ΔT is the change in temperature.
Since the iron rod's original length is 1.58 meters and the change in temperature is from 0 degrees Celsius to an unknown temperature, we can express the change in length of the iron rod as:
ΔL_iron = α_iron * L_iron * ΔT
Now, let's find the change in length of the brass rod at the same temperature change. We know that the linear expansivity of brass is 1.9 x 10^-5 K^-5.
Using the same formula as before for linear expansion, we can express the change in length of the brass rod as:
ΔL_brass = α_brass * L_brass * ΔT
Since we want the difference in lengths of the two rods to remain the same, ΔL_iron = ΔL_brass.
Now, we can set up the equation:
α_iron * L_iron * ΔT = α_brass * L_brass * ΔT
Since the change in temperature (ΔT) is the same for both rods, we can cancel it out from both sides of the equation.
α_iron * L_iron = α_brass * L_brass
Now, we can substitute the values we know into this equation:
(1.2 x 10^-5) * 1.58 = (1.9 x 10^-5) * L_brass
Simplifying this equation:
0.00001896 = (1.9 x 10^-5) * L_brass
Dividing both sides by (1.9 x 10^-5):
L_brass = 0.00001896 / (1.9 x 10^-5)
Calculating the right side of the equation:
L_brass ≈ 0.9979 meters
Therefore, the length of the brass rod at 0 degrees Celsius should be approximately 0.9979 meters to maintain the same difference in length with the iron rod at different temperatures.