To find the maximum possible error in the measurement of Young's modulus using the given data, we first need to understand the formula for Young's modulus (E), which is defined as the ratio of stress to strain. The formula is given by:
Understanding Young's Modulus
Young's modulus is expressed mathematically as:
E = (F/A) / (ΔL/L)
Where:
- E = Young's modulus
- F = applied force (in Newtons)
- A = cross-sectional area of the wire (in square centimeters)
- ΔL = extension in length (in centimeters)
- L = original length of the wire (in centimeters)
Calculating the Values
From the problem, we have:
- Diameter of the wire (d) = 0.05 cm
- Length of the wire (L) = 110 cm
- Applied load (F) = 50 N
- Extension (ΔL) = 0.125 cm
First, we calculate the cross-sectional area (A) of the wire using the formula for the area of a circle:
A = π(d/2)²
Substituting the diameter:
A = π(0.05 cm / 2)² = π(0.025 cm)² ≈ 1.9635 × 10^-3 cm²
Calculating Young's Modulus
Now we can substitute these values into the Young's modulus formula:
E = (50 N / 1.9635 × 10^-3 cm²) / (0.125 cm / 110 cm)
This simplifies to:
E = (50 / 1.9635 × 10^-3) / (0.125 / 110)
Finding Maximum Possible Error
To find the maximum possible error in Young's modulus, we need to consider the errors in each measurement. The maximum possible error can be calculated using the formula for propagation of uncertainty:
ΔE/E = (ΔF/F) + (ΔA/A) + (ΔL/L) + (ΔΔL/ΔL)
Calculating Individual Errors
Now, let's calculate the relative errors for each measurement:
- Force (F): The least count is not given, but we can assume a reasonable error, say ΔF = 1 N. Thus, ΔF/F = 1/50 = 0.02.
- Area (A): The diameter has a least count of 0.001 cm. The error in diameter (Δd) can be calculated as ΔA/A. The area error can be derived from the formula for area:
ΔA = 2π(d/2)(Δd/2) = π(0.025)(0.001) ≈ 7.85 × 10^-6 cm².
Thus, ΔA/A = (7.85 × 10^-6 / 1.9635 × 10^-3) ≈ 0.004.
- Length (L): The least count is 0.1 cm, so ΔL/L = 0.1/110 ≈ 0.000909.
- Extension (ΔL): The least count is 0.001 cm, so ΔΔL/ΔL = 0.001/0.125 = 0.008.
Combining the Errors
Now, we can sum these relative errors:
ΔE/E = 0.02 + 0.004 + 0.000909 + 0.008 ≈ 0.032909.
Calculating the Absolute Error in Young's Modulus
Finally, to find the maximum possible error in Young's modulus (ΔE), we multiply the relative error by the calculated value of E:
ΔE = ΔE/E × E.
After calculating E from the earlier steps, substitute that value here to find ΔE. This will give you the maximum possible error in your measurement of Young's modulus.
In summary, the process involves calculating the Young's modulus using the given measurements and then determining the maximum possible error by considering the uncertainties in each measurement. This approach ensures that you account for all potential sources of error in your final result.
