The different in length between a brass and iron rod is 14cm at 10 degree centgrate. What must be the length of the iron for this difference remain at 14cm when both rods are heated to 100degree
Answer
To find the length of the iron rod when both rods are heated to 100 degrees, we can use the concept of thermal expansion.
Let's define the initial length of the iron rod as L1 and the initial length of the brass rod as L2. We are given that the difference in length between the two rods at 10 degrees is 14 cm. We need to find the length of the iron rod, which we'll call L3, when the difference in length remains at 14 cm at 100 degrees.
The formula for linear thermal expansion is:
ΔL = α * L * ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L is the original length
ΔT is the change in temperature
We can rewrite the formula to solve for the change in temperature:
ΔT = ΔL / (α * L)
Since we want the difference in length to remain the same at both temperatures, we can set up an equation:
(L3 - L2) / (L1 - L2) = ΔT3 / ΔT1
Let's now plug in the given values and solve for L3:
L1 = initial length of the iron rod
L2 = initial length of the brass rod
ΔL = difference in length at 10 degrees = 14 cm
ΔT1 = change in temperature from 10 degrees
ΔT3 = change in temperature from 100 degrees
From the information given in the question, there is no mention of the coefficient of linear expansion for brass and iron. Without that information, we cannot calculate the exact length of the iron rod at 100 degrees.
To solve this problem, we need to use the concept of thermal expansion. The difference in length between the brass and iron rods at 10 degrees celsius is given as 14cm. We need to find the length of the iron rod when both rods are heated to 100 degrees celsius, while maintaining the 14cm difference in length.
The equation for thermal expansion is:
ΔL = L * α * ΔT
Where:
ΔL is the change in length
L is the original length
α is the coefficient of linear expansion
ΔT is the change in temperature
First, let's find the coefficient of linear expansion (α) for both brass and iron. The coefficient of linear expansion represents how much a material expands or contracts for each degree of temperature change.
The coefficient of linear expansion for brass is typically around 19 x 10^-6 per degree Celsius (19 * 10^-6/°C).
The coefficient of linear expansion for iron is typically around 12 x 10^-6 per degree Celsius (12 * 10^-6/°C).
Now, let's calculate the change in length (ΔL) for the brass rod when heated from 10°C to 100°C:
ΔL_brass = L_brass * α_brass * ΔT
ΔL_brass = L_brass * (19 * 10^-6/°C) * (100 - 10)
Similarly, let's calculate the change in length (ΔL) for the iron rod when heated from 10°C to 100°C:
ΔL_iron = L_iron * α_iron * ΔT
ΔL_iron = L_iron * (12 * 10^-6/°C) * (100 - 10)
Now, let's set up an equation using the given information: the difference in length between the two rods is 14cm when at 10°C.
ΔL_iron - ΔL_brass = 14 cm
Substituting the calculated values:
L_iron * (12 * 10^-6/°C) * (100-10) - L_brass * (19 * 10^-6/°C) * (100-10) = 14 cm
Since we want the difference in length to remain at 14cm when both rods are heated to 100°C, we can solve this equation to find the length of the iron rod (L_iron).