A steel tap measures the length of a copper rod as 0.8m when both are at 0^0C. What is the length of copper as measured by the steel tap when both are at 20^0C?

To solve the problem of how the length of a copper rod changes when the temperature increases from 0°C to 20°C, we need to consider the concept of thermal expansion. Both the copper rod and the steel tap will expand with an increase in temperature, but they will do so at different rates due to their distinct coefficients of linear expansion.

Understanding Thermal Expansion

Thermal expansion refers to the way materials change in size or volume in response to temperature changes. The formula for linear expansion is given by:

L = L0 + L0 * α * ΔT

Where:

• L = final length after expansion
• L0 = original length
• α = coefficient of linear expansion of the material
• ΔT = change in temperature (final temperature - initial temperature)

Coefficients of Linear Expansion

For our calculation, we need the coefficients of linear expansion for both copper and steel:

• Copper (Cu): approximately 16 x 10^-6 /°C
• Steel: approximately 11 x 10^-6 /°C

Calculating the Length of Copper at 20°C

Initially, the steel tap measures the copper rod as 0.8 m at 0°C. We need to find out how much the copper rod expands when both materials are heated to 20°C.

Step 1: Calculate the Change in Temperature

The change in temperature (ΔT) is:

ΔT = 20°C - 0°C = 20°C

Step 2: Calculate the Expansion of the Copper Rod

Using the formula for linear expansion for the copper rod:

L_cu = L0_cu + L0_cu * α_cu * ΔT

Here, L0_cu = 0.8 m, α_cu = 16 x 10^-6 /°C, and ΔT = 20°C.

Now, substituting the values:

L_cu = 0.8 m + (0.8 m * 16 x 10^-6 /°C * 20°C)

L_cu = 0.8 m + (0.8 m * 0.00032)

L_cu = 0.8 m + 0.000256 m

L_cu = 0.800256 m

Step 3: Calculate the Expansion of the Steel Tap

Next, we need to calculate how much the steel tap expands:

L_steel = L0_steel + L0_steel * α_steel * ΔT

Assuming the original length of the steel tap is also 0.8 m:

L_steel = 0.8 m + (0.8 m * 11 x 10^-6 /°C * 20°C)

L_steel = 0.8 m + (0.8 m * 0.00022)

L_steel = 0.8 m + 0.000176 m

L_steel = 0.800176 m

Final Measurement of the Copper Rod

Now, we need to find out how the length of the copper rod appears when measured with the steel tap at 20°C. Since the steel tap has also expanded, we will use the expanded length of the steel tap to measure the copper rod:

The length of the copper rod as measured by the steel tap is:

Length of copper as measured by steel tap = Length of copper / Length of steel tap

Using the expanded lengths:

Length of copper as measured = 0.800256 m / 0.800176 m * 0.8 m

Calculating this gives:

Length of copper as measured = 0.800256 m / 0.800176 m * 0.8 m ≈ 0.8001 m

Thus, the length of the copper rod as measured by the steel tap when both are at 20°C is approximately 0.8001 m. This demonstrates how both materials expand, but the measurement remains consistent due to their relative expansion rates.