The difference in length between a brass and an irod is 14cm at 10°C . What must be the length of the iron for this difference to remain at 14cm when both rods are heated to 100°C? Given that the linear expansivity of brass =19×10^-6K^-1 and iron =12×10^-6K^-1
To find the length of the iron rod when both the brass and iron rods are heated to 100°C, we can use the formula for linear expansion:
ΔL = L0 * α * ΔT
Where:
ΔL is the change in length
L0 is the initial length
α is the linear expansivity
ΔT is the change in temperature
In this case, we need to find the length of the iron rod, so we can set up the equation as follows:
ΔL_iron = L0_brass * α_brass * ΔT + L0_iron * α_iron * ΔT
Given that the difference in length between the brass and iron rods at 10°C is 14 cm, we can substitute the values into the equation:
14 cm = L0_brass * α_brass * (100°C - 10°C) + L0_iron * α_iron * (100°C - 10°C)
Since we want the difference in length to remain at 14 cm when both rods are heated to 100°C, the change in temperature is the same for both rods (100°C - 10°C), and we can simplify the equation:
14 cm = L0_brass * α_brass + L0_iron * α_iron
Now, we can solve for the length of the iron rod (L0_iron):
L0_iron = (14 cm - L0_brass * α_brass) / α_iron
Substituting the given values:
L0_iron = (14 cm - L0_brass * 19 × 10^-6 K^-1) / (12 × 10^-6 K^-1)
Note: Make sure to convert the linear expansivity values to the same units (cm and K) for consistent calculations.
Therefore, the length of the iron rod for the difference to remain at 14 cm when both rods are heated to 100°C can be obtained by plugging in the values for L0_brass, α_brass, and α_iron into the equation above.