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Calculate the density of a fiber-reinforced polymer composite material based on measurements of the sample's size, weight in water, and known uncertainties. Analyze how measurement uncertainties contribute to the overall uncertainty in density determination, and propose improvements to reduce the uncertainty to acceptable levels.

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Determining the density of a fiber-reinforced polymer (FRP) composite material is essential for understanding its mechanical properties, performance, and suitability for various engineering applications. The experimental approach involves measuring the sample's dimensions, its weight when submerged in water, and subsequently calculating the density using Archimedes' principle. The challenge resides in accurately quantifying how measurement uncertainties influence the final calculation of density, and how to optimize measurement procedures for increased precision.

Introduction

Density is a fundamental property of materials that indicates mass per unit volume, directly influencing their strength, durability, and application range. For composite materials like fiber-reinforced polymers, precise density measurements inform engineers about the material's quality and consistency, impacting product performance. The experimental measurement typically involves dimensional assessments, mass measurement, and water displacement techniques, which are susceptible to uncertainties that propagate through to the density calculation.

Methodology and Calculation

Given the nominal dimensions of the composite sample: 4 mm in width, 5 mm in length, and 1 mm in thickness, the volume \(V\) can be calculated. The sample's weight in water (\(W_w\)) is 16 mg, which allows the determination of its buoyant volume. The known density of water (\(\rho_{water}\)) is 1000 kg/m\(^3\). First, convert all measurements to SI units for consistency:

  • Dimensions: \(4\,\text{mm} = 4 \times 10^{-3}\,\text{m}\), \(5\,\text{mm} = 5 \times 10^{-3}\,\text{m}\), \(1\,\text{mm} = 1 \times 10^{-3}\,\text{m}\)
  • Mass in water: 16 mg = 16 \(\times\) 10\(^{-6}\) kg

The volume of the sample, determined via water displacement, is given by the ratio of buoyant force to water density. Using Archimedes' principle:

\(V = \dfrac{W_{in\ water}}{\rho_{water}}\)

Calculating the sample's density \(\rho_{sample}\):

\(\rho_{sample} = \dfrac{mass}{volume} = \dfrac{16 \times 10^{-6}\, \text{kg}}{V}\)

Since the volume \(V\) is derived from the water displacement method, it accounts for the sample's actual volume, including any potential measurement uncertainties.

Part 1: Nominal Density Calculation

The nominal volume based on dimensions is:

\(V_{nominal} = 4 \times 10^{-3} \text{ m} \times 5 \times 10^{-3} \text{ m} \times 1 \times 10^{-3} \text{ m} = 2 \times 10^{-8} \text{ m}^3\)

The mass in water is 16 mg, so the sample's apparent volume in water (assuming negligible water absorption) is:

\(V_{measured} = \dfrac{mass\, in\, water}{\rho_{water}} = \dfrac{16 \times 10^{-6}\, \text{kg}}{1000\, \text{kg/m}^3} = 1.6 \times 10^{-8}\, \text{m}^3\)

The approximate density of the composite material is then:

\(\rho_{composite} = \dfrac{16 \times 10^{-6}\, \text{kg}}{1.6 \times 10^{-8}\, \text{m}^3} = 1000\, \text{kg/m}^3\)

which matches the water density, indicating the need for precise volume measurement and correction for measurement uncertainties.

Part 2: Uncertainty Analysis (Uncertainties on Measurements)

The uncertainties on measurements are given as:

  • Scale (mass): \(\pm 0.2\, \text{mg} = \pm 2 \times 10^{-7}\, \text{kg}\)
  • Dimension measurement: \(\pm 0.25\, \text{mm} = \pm 0.25 \times 10^{-3}\, \text{m}\)
  • Density of water: 1% uncertainty

The total uncertainty on the density is propagated from these measurements using standard error propagation formulas. The relative uncertainty components are calculated as:

  • Mass uncertainty: \(\frac{0.2\, \text{mg}}{16\, \text{mg}} = 1.25\%\)
  • Dimension uncertainty affects volume calculations; with uncertainty in each length dimension (assuming independent errors), the combined relative uncertainty in volume is approximately:

\(\delta V / V = \sqrt{ (\delta L / L)^2 + (\delta W / W)^2 + (\delta T / T)^2 } = \sqrt{ (0.25/4)^2 + (0.25/5)^2 + (0.25/1)^2 } \approx 25\%\)

Although the simplified calculation indicates a significant impact from dimensional uncertainty, more refined analysis considers the errors in volume directly. The uncertainty in volume due to dimension errors leads to a significant contribution to density uncertainty.

The final density uncertainty combines mass and volume uncertainties, as well as the water density uncertainty:

\(\delta \rho / \rho = \sqrt{ (\delta m / m)^2 + (\delta V / V)^2 + (\delta \rho_{water} / \rho_{water})^2 }\)

Plugging in the values:

\(\delta \rho / \rho \approx \sqrt{ (1.25\%)^2 + (25\%)^2 + (1\%)^2 } \approx 25.1\%\)

Part 3: Reduced Dimension Uncertainty (0.1 mm)

If the measurement uncertainty in dimensions decreases to 0.1 mm, the relative error in each dimension is halved, leading to:

\(\delta V / V \approx \sqrt{ (0.1/4)^2 + (0.1/5)^2 + (0.1/1)^2 } \approx 12.5\%\)

Consequently, the total uncertainty in density reduces significantly:

\(\delta \rho / \rho \approx \sqrt{ (1.25\%)^2 + (12.5\%)^2 + (1\%)^2 } \approx 12.6\%\)

Part 4: Improving Procedure for 5% or Less Uncertainty

To reduce combined measurement uncertainty to below 5%, several procedural enhancements are recommended:

  • Utilize more precise measurement tools, such as digital micrometers with ±0.01 mm accuracy, to minimize dimensional errors.
  • Employ high-precision electronic scales with at least ±0.01 mg resolution for mass measurement.
  • Perform multiple measurements and average results to mitigate random errors.
  • Ensure environmental stability (temperature, vibrations) during measurements to prevent measurement drift.
  • Implement calibration protocols for measurement instruments regularly.
  • Use better water displacement techniques, such as a specialized pycnometer, to improve volume accuracy.

By implementing these improvements, it is feasible to achieve an overall density measurement uncertainty below 5%, making the data more reliable for engineering applications.

Part 5: Using a Micrometer to Reduce Uncertainty in Thickness

Using a micrometer accurate to ±1 mil (approximately ±0.0254 mm) for dimension measurement introduces greater precision. If the scales on other measurements have similar or better accuracy, reducing the sample thickness to less than 1 mm with this resolution is possible.

Reducing the thickness would decrease the volume uncertainty and, consequently, the uncertainty in the density calculation. Given the original uncertainties and assuming the micrometer's accuracy is maintained, the combined uncertainty could indeed be less than 2%.

However, practical challenges include handling smaller samples and ensuring measurement repeatability at such fine scales. Any slight misalignment or measurement error could reduce the benefits. Still, with high-quality measurement equipment and strict measurement protocols, achieving less than 2% uncertainty in density measurement is attainable.

In conclusion, selecting precise measurement tools like a micrometer for sample dimensions enhances the likelihood of reducing the overall density uncertainty to below 2%. Ensuring proper calibration and measurement techniques is essential for realizing this precision in practice.

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