1...At what temp. of 0^C steel rod and copper rod have length of 800.1 & 99.9 respectively. At what temp. will the 2 rods have the same length.
coeff.of linear expansion Copper-.000017 steel-.000013
2...The length of a brass bar increase from 4.001 into 4.0030 m. find the final temperature of the bar if the initial temperature was 0 Celsius.
coeff. of linear expansion brass-.000019
To find the temperature at which the steel rod and copper rod have the same length, we can use the concept of linear expansion. The equation for linear expansion is given by:
ΔL = αL0ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the initial length of the rod
ΔT is the change in temperature
For the first question:
Let's assume the initial temperature of the rods is T0.
For the steel rod, the initial length is 800.1 and the coefficient of linear expansion is 0.000013.
For the copper rod, the initial length is 99.9 and the coefficient of linear expansion is 0.000017.
At the temperature at which the two rods have the same length, the change in length for both rods will be equal. So we can set up an equation:
0.000013 * 800.1 * (T - T0) = 0.000017 * 99.9 * (T - T0)
Simplifying the equation:
0.0104013 * (T - T0) = 0.0016983 * (T - T0)
Now, we can solve for T by canceling out (T - T0):
0.0104013 = 0.0016983
T - T0 = 0
The two equations are equal. Therefore, the rods will have the same length at any temperature T.
For the second question:
We can use the same equation for linear expansion, ΔL = αL0ΔT.
Let's assume the initial temperature of the brass bar is T0 and the change in temperature is ΔT.
ΔL = 0.000019 * 4.001 * ΔT
Since the change in length is given as 0.0020, we can substitute this value:
0.0020 = 0.000019 * 4.001 * ΔT
Now, we can solve for ΔT:
ΔT = 0.0020 / (0.000019 * 4.001)
Calculating ΔT:
ΔT = 0.0020 / 0.000076019
ΔT ≈ 26.302
The final temperature of the brass bar would be the initial temperature (0 Celsius) plus ΔT:
Final temperature = 0 + 26.302
The final temperature of the brass bar would be approximately 26.302 Celsius.