A steel ruler is calibrated to read true at 21.6 °C. A draftsman uses the ruler at 44.5 °C to draw a line on a 44.5 °C copper plate. As indicated on the warm ruler, the length of the line is 0.314 m. To what temperature should the plate be cooled, such that the length of the line truly becomes 0.314 m?
To solve this problem, we need to use the concept of thermal expansion. The length of an object changes with temperature due to thermal expansion.
The first step is to calculate the coefficient of linear expansion for both steel and copper. The equation for thermal expansion is:
ΔL = α * L * ΔT
Where:
ΔL = change in length
α = coefficient of linear expansion
L = original length
ΔT = change in temperature
The coefficient of linear expansion for steel is approximately 12 x 10^-6 °C^-1, and for copper, it is approximately 16 x 10^-6 °C^-1.
Let's assume the final temperature of the copper plate is T. At this temperature, the change in length for the steel ruler should be zero, as it should read true. Thus, the change in length for the line drawn on the copper plate should also be zero.
Using the equation for thermal expansion, we can set up the following equation:
0 = α_steel * L * (T - 21.6) - α_copper * L * (T - 44.5)
Now we can solve for T:
α_steel * L * (T - 21.6) = α_copper * L * (T - 44.5)
(T - 21.6) / (T - 44.5) = α_copper / α_steel
[(T - 21.6) / (T - 44.5)] = (16 x 10^-6 °C^-1) / (12 x 10^-6 °C^-1)
Simplifying the equation:
(T - 21.6) / (T - 44.5) = 4 / 3
Cross-multiplying:
3(T - 21.6) = 4(T - 44.5)
3T - 64.8 = 4T - 178
Transposing terms:
4T - 3T = 178 - 64.8
T = 113.2 °C
Therefore, the copper plate should be cooled to 113.2 °C in order for the length of the line to be truly 0.314 m when measured with the steel ruler.
To solve this problem, we can use the concept of thermal expansion. When an object is heated or cooled, it expands or contracts in size. In this case, both the ruler and the copper plate will undergo thermal expansion.
The first step is to determine the initial length of the line on the ruler at 21.6 °C and the final length required at the same temperature. Let's call the initial length L1 and the final length L2.
Given:
- Initial temperature of the ruler = 21.6 °C
- Final temperature of the ruler = 44.5 °C
- Length of the line at 44.5 °C = 0.314 m
We need to find:
- The temperature at which the copper plate needs to be cooled (let's call it T)
To find L1, we can use the concept that the change in length of an object is proportional to the change in temperature. This can be expressed as:
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
In this case, we'll use the ruler's coefficient of linear expansion, assuming it remains constant over the given temperature range. The coefficient of linear expansion for steel is typically around 12 x 10^-6 °C^-1.
Let's calculate the initial length (L1) of the line on the ruler at 21.6 °C using the above formula:
ΔL1 = α * L1 * ΔT1
L1 = ΔL1 / (α * ΔT1)
= 0.314 m / (12 x 10^-6 °C^-1 * (44.5 °C - 21.6 °C))
Now, we can calculate L1.
Next, we need to find the change in length (ΔL2) required to make the length of the line 0.314 m at 21.6 °C. Since we want this to be the same length as L1, ΔL2 = 0.
Finally, we'll calculate the change in temperature (ΔT2) of the copper plate required for the length of the line to remain the same at 21.6 °C. Again, using the formula: ΔL = α * L * ΔT, we'll find ΔT2.
Once we have ΔT2, we can calculate the required temperature (T) by subtracting ΔT2 from the initial temperature of the plate (44.5 °C).
Following this process, you should be able to find the temperature at which the copper plate needs to be cooled such that the length of the line becomes 0.314 m.