Mass Conversion 83D245

1. **State the problem:** We have five masses with different units and need to find the total mass in kilograms, pounds, and slugs. 2. **Given data:** - $m_1 = 500$ g - $m_2 = 800$ gf (gram-force) - $m_3 = 15$ poundals - $m_4 = 3$ lbf (pound-force) - $m_5 = 0.10$ slug - Acceleration due to gravity $g = 30.5$ ft/s² 3. **Important conversions and formulas:** - $1$ slug $= 32.174$ lbm (pounds mass) - $1$ lbm $= 0.453592$ kg - $1$ g $= 0.001$ kg - $1$ lbf $= 1$ lbm * $g_{standard}$ (standard gravity $= 32.174$ ft/s²) - $1$ gf $= 1$ g * $g_{standard}$ (standard gravity $= 9.80665$ m/s²) but here gravity is $30.5$ ft/s², so we adjust accordingly - Poundal is defined as $1$ poundal $= 1$ lbm * 1 ft/s² 4. **Calculate each mass in pounds mass (lbm):** - $m_1$: Convert grams to kilograms then to pounds mass $$m_1 = 500 \text{ g} = 500 \times 0.001 = 0.5 \text{ kg}$$ $$0.5 \text{ kg} \times \frac{1}{0.453592} = 1.10231 \text{ lbm}$$ - $m_2$: Given as 800 gf. Since $1$ gf $= 1$ g * $g_{standard}$, but gravity here is $30.5$ ft/s² instead of $9.80665$ m/s². First convert $g$ to ft/s²: $9.80665$ m/s² $= 32.174$ ft/s² (standard gravity). Here gravity is $30.5$ ft/s², so force in gf is proportional to $\frac{30.5}{32.174}$ of standard. Force in pounds force: $$800 \text{ gf} = 800 \times \frac{30.5}{32.174} \text{ gf (adjusted)} = 758.7 \text{ gf (adjusted)}$$ Since $1$ lbf $= 32.174$ lbm*ft/s², and $1$ gf $= 1$ g * $g_{standard}$, to get mass in lbm: $$m_2 = \frac{758.7 \text{ gf}}{30.5 \text{ ft/s}^2} = 24.88 \text{ lbm}$$ - $m_3$: 15 poundals. Since $1$ poundal $= 1$ lbm * 1 ft/s², mass in lbm is force divided by acceleration: $$m_3 = \frac{15 \text{ poundals}}{30.5 \text{ ft/s}^2} = 0.4918 \text{ lbm}$$ - $m_4$: 3 lbf. Using $g = 30.5$ ft/s², mass in lbm is force divided by acceleration: $$m_4 = \frac{3 \text{ lbf}}{30.5 \text{ ft/s}^2} = 0.09836 \text{ lbm}$$ - $m_5$: Already in slugs, convert to lbm: $$m_5 = 0.10 \text{ slug} \times 32.174 = 3.2174 \text{ lbm}$$ 5. **Sum all masses in lbm:** $$m_{total} = 1.10231 + 24.88 + 0.4918 + 0.09836 + 3.2174 = 29.79 \text{ lbm}$$ 6. **Convert total mass to kilograms:** $$m_{total,kg} = 29.79 \times 0.453592 = 13.52 \text{ kg}$$ 7. **Convert total mass to slugs:** $$m_{total,slug} = \frac{29.79}{32.174} = 0.926 \text{ slug}$$ **Final answers:** - (a) Total mass in kilograms: $13.52$ kg - (b) Total mass in pounds: $29.79$ lbm - Total mass in slugs: $0.926$ slug