the length of an aluminum is 76.5cm and that of an iron ruler is 80cm at 0 degree celsius at what temperature will the length of the rules become
lengthAl=alengthFe
76.5(1+coeffAl(deltaTemp))=80(1+coeffFe(deltatemp)
76.5/80 (1+coeffAl*deltaT)= (1+coeffFe(deltatemp)
76.5/80 +76.5/80 *coeffAl*deltaT)= 1+coeffFe(deltatemp)
deltaTemp (76.5/80*coeffAl-coeffFe)=1-76.5/80
solve for delta Temp
check my math.
To find the temperature at which the length of the rulers will become equal, we can use the concept of thermal expansion.
The formula for linear expansion is given by:
ΔL = α * L0 * ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the initial length of the object
ΔT is the change in temperature
In this case, we want to find the temperature at which the lengths of the rulers will be equal. We can set up the equation as follows:
ΔL1 = α1 * L01 * ΔT
ΔL2 = α2 * L02 * ΔT
Where:
ΔL1 = ΔL2 (the change in length is the same for both rulers)
L01 is the initial length of the aluminum ruler (76.5 cm)
L02 is the initial length of the iron ruler (80 cm)
α1 is the coefficient of linear expansion for aluminum
α2 is the coefficient of linear expansion for iron
Rearranging the equations, we get:
ΔT = ΔL1 / (α1 * L01) = ΔL2 / (α2 * L02)
Substituting the given values:
L01 = 76.5 cm
L02 = 80 cm
We need the coefficients of linear expansion for aluminum and iron. The coefficient of linear expansion for aluminum is approximately 0.000022 cm/°C and for iron is approximately 0.000012 cm/°C.
Substituting the values, we get:
ΔT = ΔL1 / (0.000022 * 76.5) = ΔL2 / (0.000012 * 80)
Simplifying the equation:
ΔT = ΔL1 / 0.00143 = ΔL2 / 0.00096
Since ΔL1 = ΔL2, we can further simplify the equation:
ΔT = ΔL1 / 0.00143 = ΔL1 / 0.00096
Cross multiplying:
0.00096 * ΔL1 = 0.00143 * ΔL1
Canceling ΔL1, we get:
0.00096 = 0.00143
This is not possible. It means that the lengths of both rulers at any temperature will never be equal.
Therefore, there is no specific temperature at which the lengths of the rulers become equal.
To find the temperature at which the length of the rulers become equal, we can use the formula for thermal expansion:
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
In this case, we want to find the temperature at which the lengths are equal, so ΔL = 0. Therefore, we can set up the equation:
0 = L_aluminum * α_aluminum * ΔT - L_iron * α_iron * ΔT
Simplifying, we can divide both sides by ΔT:
0 = L_aluminum * α_aluminum - L_iron * α_iron
Now we can solve for the temperature by rearranging the equation:
α_aluminum * L_aluminum = α_iron * L_iron
We have the following values:
L_aluminum = 76.5 cm
L_iron = 80 cm
Now, we need to find the coefficients of linear expansion for aluminum and iron. The coefficient of linear expansion (α) is a material-specific constant. For aluminum, the coefficient of linear expansion is approximately 0.000022/°C, and for iron, it is approximately 0.000012/°C.
Substituting the known values:
α_aluminum * 76.5 = α_iron * 80
Now we can solve for the unknown temperature:
α_aluminum / α_iron = 80 / 76.5
α_aluminum / α_iron ≈ 1.0458
Solving for α_aluminum / α_iron, we find that the ratio is approximately 1.0458.
Finally, to find the temperature at which the lengths of the rulers become equal, we need to divide the coefficients of linear expansion for aluminum and iron:
0.000022 / 0.000012 ≈ 1.8333
This means that the lengths of the rulers will be equal at approximately 1.8333 times the coefficient of linear expansion of aluminum, or approximately 1.8333 times the coefficient of linear expansion of iron.
Therefore, to find the temperature at which the lengths of the rulers become equal, you need to multiply the coefficient of linear expansion of aluminum by 1.8333.