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Determine the target center of gravity (CG) position.

Physics

ScanSolve AI Answer

Answer

D

Step-by-step solution

6 steps

To solve this problem, we need to use the principle of moments. When mass is removed, the total moment of the aircraft changes, which in turn shifts the center of gravity (CG).

  1. Determine the target center of gravity (CG) position.

    The CG limits are between 18 m and 22 m. The middle of these limits is:

    CGtarget=18 m+22 m2=40 m2=20 mCG_{target} = \frac{18 \text{ m} + 22 \text{ m}}{2} = \frac{40 \text{ m}}{2} = 20 \text{ m}
  2. Set up the moment equation.

    The initial moment of the aircraft is Minitial×CGinitialM_{initial} \times CG_{initial}.
    When a mass mm is removed from position XremovedX_{removed}, the new total mass is Minitial−mM_{initial} - m, and the new CG is CGtargetCG_{target}.
    The moment equation is:

    (Minitial−m)×CGtarget=(Minitial×CGinitial)−(m×Xremoved)(M_{initial} - m) \times CG_{target} = (M_{initial} \times CG_{initial}) - (m \times X_{removed})

    Given values:
    Minitial=112000 kgM_{initial} = 112000 \text{ kg}
    CGinitial=22.62 mCG_{initial} = 22.62 \text{ m}
    CGtarget=20 mCG_{target} = 20 \text{ m}
    Xremoved=30 mX_{removed} = 30 \text{ m}

  3. Substitute the values into the equation.

    (112000 kg−m)×20 m=(112000 kg×22.62 m)−(m×30 m)(112000 \text{ kg} - m) \times 20 \text{ m} = (112000 \text{ kg} \times 22.62 \text{ m}) - (m \times 30 \text{ m})
  4. Expand and simplify the equation.

    (112000×20)−(20m)=(112000×22.62)−(30m)(112000 \times 20) - (20m) = (112000 \times 22.62) - (30m) 2240000−20m=2533440−30m2240000 - 20m = 2533440 - 30m
  5. Isolate the term with mm.

    Add 30m30m to both sides:

    2240000+10m=25334402240000 + 10m = 2533440

    Subtract 2240000 from both sides:

    10m=2533440−224000010m = 2533440 - 2240000 10m=29344010m = 293440
  6. Solve for mm.

    m=29344010m = \frac{293440}{10} m=29344 kgm = 29344 \text{ kg}

    The mass that must be removed is 29 344 kg.

    Comparing this to the given options:
    A) 8 680 kg.
    B) 43 120 kg.
    C) 16 529 kg.
    D) 29 344 kg.

    The correct option is D.

    The mass that must be removed is:

    D) 29 344 kg.\boxed{\text{D) 29 344 kg.}}
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