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If the mass of the moon and its radius are 7.40 x 1022 kg and 1.76 x 106 m respectively, find the free fall acceleration g of an object on or near the surface of the moon.

Physics

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Answer

1.59 m/s2\text{1.59 m/s}^2

Step-by-step solution

6 steps

  1. Identify the formula for free fall acceleration.

    The free fall acceleration gg on the surface of a celestial body is given by the formula:

    g=GMR2g = \frac{GM}{R^2}

    where GG is the gravitational constant, MM is the mass of the celestial body, and RR is its radius.

  2. List the given values and the gravitational constant.

    • Mass of the moon, M=7.40×1022 kgM = 7.40 \times 10^{22} \text{ kg}
    • Radius of the moon, R=1.76×106 mR = 1.76 \times 10^6 \text{ m}
    • Gravitational constant, G=6.674×10−11 N⋅m2/kg2G = 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2

  3. Substitute the values into the formula.

    g=(6.674×10−11 N⋅m2/kg2)×(7.40×1022 kg)(1.76×106 m)2g = \frac{(6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2) \times (7.40 \times 10^{22} \text{ kg})}{(1.76 \times 10^6 \text{ m})^2}

  4. Calculate the square of the radius.

    (1.76×106 m)2=(1.76)2×(106)2 m2=3.0976×1012 m2(1.76 \times 10^6 \text{ m})^2 = (1.76)^2 \times (10^6)^2 \text{ m}^2 = 3.0976 \times 10^{12} \text{ m}^2

  5. Perform the multiplication in the numerator.

    (6.674×10−11)×(7.40×1022)=(6.674×7.40)×(10−11×1022)(6.674 \times 10^{-11}) \times (7.40 \times 10^{22}) = (6.674 \times 7.40) \times (10^{-11} \times 10^{22}) =49.3876×1011 N⋅m2/kg= 49.3876 \times 10^{11} \text{ N} \cdot \text{m}^2/\text{kg}

  6. Divide the numerator by the squared radius.

    g=49.3876×1011 N⋅m2/kg3.0976×1012 m2g = \frac{49.3876 \times 10^{11} \text{ N} \cdot \text{m}^2/\text{kg}}{3.0976 \times 10^{12} \text{ m}^2} g=49.38763.0976×10111012 m/s2g = \frac{49.3876}{3.0976} \times \frac{10^{11}}{10^{12}} \text{ m/s}^2 g≈15.943×10−1 m/s2g \approx 15.943 \times 10^{-1} \text{ m/s}^2 g≈1.5943 m/s2g \approx 1.5943 \text{ m/s}^2

    The free fall acceleration on the surface of the moon is approximately:

    1.59 m/s2\text{1.59 m/s}^2
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