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To determine which bag is heavier when measured in air—the 9.99-pound bag of steel ingots or the 10.01-pound bag of fluffy cotton—we need to consider not just their masses but also the effect of air buoyancy on their apparent weights.
*Step 1: Calculate the Masses*
First, convert the given weights into masses using the conversion \(1 \text{ pound} = 0.453592 \text{ kg}\):
- Mass of steel bag, \(m_s = 9.99 \, \text{lb} \times 0.453592 \, \text{kg/lb} \approx 4.5310 \, \text{kg}\)
- Mass of cotton bag, \(m_c = 10.01 \, \text{lb} \times 0.453592 \, \text{kg/lb} \approx 4.5401 \, \text{kg}\)
*Step 2: Calculate the Volumes*
Next, calculate their volumes using the densities:
- Density of steel, \(\rho_s \approx 8000 \, \text{kg/m}^3\)
- Density of cotton (fluffy), \(\rho_c \approx 50 \, \text{kg/m}^3\)
- Volume of steel bag, \(V_s = \frac{m_s}{\rho_s} \approx \frac{4.5310 \, \text{kg}}{8000 \, \text{kg/m}^3} \approx 5.664 \times 10^{-4} \, \text{m}^3\)
- Volume of cotton bag, \(V_c = \frac{m_c}{\rho_c} \approx \frac{4.5401 \, \text{kg}}{50 \, \text{kg/m}^3} \approx 0.090802 \, \text{m}^3\)
*Step 3: Calculate the Buoyant Forces*
Using the density of air \(\rho_{\text{air}} \approx 1.2 \, \text{kg/m}^3\) and acceleration due to gravity \(g = 9.81 \, \text{m/s}^2\):
- Buoyant force on steel bag, \(B_s = \rho_{\text{air}} \times V_s \times g \approx 1.2 \, \text{kg/m}^3 \times 5.664 \times 10^{-4} \, \text{m}^3 \times 9.81 \, \text{m/s}^2 \approx 0.00668 \, \text{N}\)
- Buoyant force on cotton bag, \(B_c = \rho_{\text{air}} \times V_c \times g \approx 1.2 \, \text{kg/m}^3 \times 0.090802 \, \text{m}^3 \times 9.81 \, \text{m/s}^2 \approx 1.068 \, \text{N}\)
*Step 4: Calculate the Apparent Weights in Air*
Subtract the buoyant force from the actual gravitational force (mass times gravity):
- Apparent weight of steel bag, \(W_s' = m_s \times g - B_s \approx 4.5310 \, \text{kg} \times 9.81 \, \text{m/s}^2 - 0.00668 \, \text{N} \approx 44.44 \, \text{N}\)
- Apparent weight of cotton bag, \(W_c' = m_c \times g - B_c \approx 4.5401 \, \text{kg} \times 9.81 \, \text{m/s}^2 - 1.068 \, \text{N} \approx 43.47 \, \text{N}\)
*Conclusion:*
Despite the cotton bag having slightly more mass and a greater weight in a vacuum, when measured in air, the steel bag is heavier due to the significantly smaller upward buoyant force acting on it compared to the cotton bag. This means that on a scale in air, the 9.99-pound bag of steel ingots will weigh more than the 10.01-pound bag of fluffy cotton.