1..........At what temperature of 0^C steel rod & copper rod have length of 800.1cm and 99.9cm respectively. At what temperature will the 2 rods have the same length.
Coefficients of linear expansion (per °C) copper rod----.000017 steel rod-------.000012
2.....The length of a brass bar increase from 4.001m into 4.0030 m. find the final temperature of the bar if the initial temperature was 0 Celsius.
Coefficients of linear expansion (per °C) brass----- .000019
1. You'd better check your numbers. If the copper rod starts out 8 times longer, no amount of heating (or cooling) is going to make it shorter than the steel rod.
Is your 800.1 cm supposed to be 100.1 cm?
To solve these problems, we can use the formula for linear expansion:
ΔL = α * L * ΔT
Where:
- ΔL is the change in length
- α is the coefficient of linear expansion
- L is the initial length
- ΔT is the change in temperature
1. At what temperature will the steel rod and copper rod have the same length?
Let's assume that the initial temperature is 0°C, and we need to find the temperature at which both rods have the same length.
For the steel rod:
ΔL_steel = α_steel * L_steel * ΔT
For the copper rod:
ΔL_copper = α_copper * L_copper * ΔT
We know that when the rods have the same length, ΔL_steel = ΔL_copper. Setting these two equations equal to each other:
α_steel * L_steel * ΔT = α_copper * L_copper * ΔT
We can cancel out ΔT:
α_steel * L_steel = α_copper * L_copper
Now we can solve for the temperature at which the rods will have the same length. Rearranging the equation:
ΔT = (α_copper * L_copper) / α_steel * L_steel
Substituting the given values:
ΔT = (0.000017 * 99.9) / (0.000012 * 800.1)
Calculating that:
ΔT ≈ 1.2734°C
Therefore, the two rods will have the same length at approximately 1.2734°C.
2. Find the final temperature of the brass bar if the initial temperature was 0°C.
Given:
ΔL = 4.0030 m - 4.0010 m = 0.0020 m
α_brass = 0.000019
We need to find ΔT using the formula ΔL = α * L * ΔT and solve for ΔT:
ΔT = ΔL / (α_brass * L)
Substituting the given values:
ΔT = 0.0020 / (0.000019 * 4.0010)
Calculating that:
ΔT ≈ 2.6329°C
Therefore, the final temperature of the brass bar will be approximately 2.6329°C.