Air at sea level has a density of 0.00238 slugs/ft^3 (1 slug = 32.174 g in pounds). What is the density in kg/m^3? (Convert g in m/s^2 to ft/s^2, then convert lb/ft^3 to kg/m^3)
The Correct Answer and Explanation is:
Step-by-Step Solution:
We are asked to convert air density from slugs/ft³ to kg/m³, following these conversions:
Given Data:
- Density = 0.00238 slugs/ft³
- 1 slug = 32.174 lb
- 1 lb = 0.45359237 kg
- 1 ft = 0.3048 m
- So, 1 ft³ = (0.3048)³ = 0.02831685 m³
Step 1: Convert slugs/ft³ to lb/ft³
We know:1 slug=32.174 lb1 \text{ slug} = 32.174 \text{ lb}1 slug=32.174 lb
Therefore:0.00238 slugs/ft3=0.00238×32.174=0.07656 lb/ft30.00238 \, \text{slugs/ft}^3 = 0.00238 \times 32.174 = 0.07656 \, \text{lb/ft}^30.00238slugs/ft3=0.00238×32.174=0.07656lb/ft3
Step 2: Convert lb/ft³ to kg/m³
We know:1 lb=0.45359237 kg1 \, \text{lb} = 0.45359237 \, \text{kg}1lb=0.45359237kg1 ft3=0.02831685 m31 \, \text{ft}^3 = 0.02831685 \, \text{m}^31ft3=0.02831685m3
Thus:0.07656 lb/ft3=0.07656×0.453592370.02831685=1.226 kg/m30.07656 \, \text{lb/ft}^3 = \frac{0.07656 \times 0.45359237}{0.02831685} = 1.226 \, \text{kg/m}^30.07656lb/ft3=0.028316850.07656×0.45359237=1.226kg/m3
Final Answer:
The density of air at sea level is approximately 1.226 kg/m³.
Explanation:
The slug is a unit of mass used in the imperial system, primarily for engineering calculations involving force and mass. To convert air density to SI units, we first translate slugs to pounds using the defined relationship between slugs and pounds of mass. Next, pounds are converted to kilograms based on the standard metric equivalence. Finally, volume is converted from cubic feet to cubic meters using the linear conversion factor for feet to meters, cubed to account for the three-dimensional measurement.
This process provides a precise way to express air density in terms consistent with international scientific standards. The final result of approximately 1.226 kg/m³ is widely accepted for air at sea level under standard conditions.
