Mass Conversion E27Df1

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 - $m_4 = 3$ lb₍f₎ (pounds-force) - $m_5 = 0.10$ slug - Acceleration due to gravity $g = 30.5$ ft/s² 3. **Important formulas and conversions:** - $1$ slug $= 32.174$ lbₘ (pounds mass) - $1$ lb₍f₎ $= 1$ pound-force $= 1$ lbₘ $\times g_{standard}$ where $g_{standard} = 32.174$ ft/s² - $1$ poundal $= 1$ lbₘ $\times 1$ ft/s² - $1$ gf $= 1$ gram-force $= 1$ g $\times 9.80665$ m/s² (standard gravity) - $1$ lbₘ $= 0.45359237$ kg - $1$ slug $= 14.5939$ kg - $1$ g $= 0.001$ kg 4. **Convert all masses to a common unit, starting with slugs:** - $m_1 = 500$ g $= 500 \times 0.001 = 0.5$ kg Convert kg to slugs: $$m_1 = \frac{0.5}{14.5939} = 0.03426 \text{ slugs}$$ - $m_2 = 800$ gf Since $1$ gf $= 1$ g $\times 9.80665$ m/s², but gravity here is $30.5$ ft/s², convert gf to lb₍f₎ first: Convert $800$ gf to Newtons: $$800 \times 9.80665 \times 10^{-3} = 7.84532 \text{ N}$$ Convert Newtons to pound-force: $$1 \text{ lbf} = 4.44822 \text{ N} \Rightarrow 7.84532 / 4.44822 = 1.763 \text{ lbf}$$ Convert lbf to slugs: $$m_2 = \frac{1.763}{30.5} = 0.05782 \text{ slugs}$$ - $m_3 = 15$ poundals Since $1$ poundal $= 1$ lbₘ $\times 1$ ft/s², mass in lbₘ: $$m_3 = \frac{15}{30.5} = 0.4918 \text{ lb}_m$$ Convert lbₘ to slugs: $$m_3 = \frac{0.4918}{32.174} = 0.01528 \text{ slugs}$$ - $m_4 = 3$ lb₍f₎ Convert lb₍f₎ to lbₘ: $$m_4 = \frac{3}{30.5} = 0.09836 \text{ lb}_m$$ Convert lbₘ to slugs: $$m_4 = \frac{0.09836}{32.174} = 0.00306 \text{ slugs}$$ - $m_5 = 0.10$ slug (already in slugs) 5. **Sum all masses in slugs:** $$m_{total} = 0.03426 + 0.05782 + 0.01528 + 0.00306 + 0.10 = 0.21042 \text{ slugs}$$ 6. **Convert total mass to kilograms:** $$m_{total,kg} = 0.21042 \times 14.5939 = 3.071 \text{ kg}$$ 7. **Convert total mass to pounds (lbₘ):** $$m_{total,lb} = 0.21042 \times 32.174 = 6.766 \text{ lb}_m$$ **Final answers:** - (a) Total mass in kilograms: $3.071$ kg - (b) Total mass in pounds: $6.766$ lbₘ - Total mass in slugs: $0.21042$ slugs