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Calculate the mass of a planet using Newton's law of universal gravitation and the acceleration due to gravity on its surface.

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Answer

3.243×1023 kg3.243 \times 10^{23} \text{ kg}

57.6 N57.6 \text{ N}

Step-by-step solution

6 steps

5.2.1 Newton's law of universal gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

5.2.2

  1. Identify the given values and constants.

    Radius of the planet, R=2.6×106 mR = 2.6 \times 10^6 \text{ m}
    Acceleration due to gravity on the planet, g=3.2 m⋅s−2g = 3.2 \text{ m}\cdot\text{s}^{-2}
    Gravitational constant, G=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2

  2. Use the formula for gravitational acceleration on the surface of a planet.

    g=GMpR2g = \frac{GM_p}{R^2}

    Rearrange the formula to solve for the mass of the planet (MpM_p).

    Mp=gR2GM_p = \frac{gR^2}{G}

  3. Substitute the values and calculate MpM_p.

    Mp=(3.2 m⋅s−2)×(2.6×106 m)26.67×10−11 N⋅m2/kg2M_p = \frac{(3.2 \text{ m}\cdot\text{s}^{-2}) \times (2.6 \times 10^6 \text{ m})^2}{6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2} Mp=3.2×(6.76×1012)6.67×10−11 kgM_p = \frac{3.2 \times (6.76 \times 10^{12})}{6.67 \times 10^{-11}} \text{ kg} Mp=2.1632×10136.67×10−11 kgM_p = \frac{2.1632 \times 10^{13}}{6.67 \times 10^{-11}} \text{ kg} Mp=3.243×1023 kgM_p = 3.243 \times 10^{23} \text{ kg}

    The mass of the planet is 3.243×1023 kg\boxed{3.243 \times 10^{23} \text{ kg}}.

    5.2.3

  4. Identify the given values.

    Mass of the object, m=18 kgm = 18 \text{ kg}
    Acceleration due to gravity on the planet, g=3.2 m⋅s−2g = 3.2 \text{ m}\cdot\text{s}^{-2} (from the problem statement)

  5. Use the formula for weight.

    W=mgW = mg

  6. Substitute the values and calculate the weight.

    W=(18 kg)×(3.2 m⋅s−2)W = (18 \text{ kg}) \times (3.2 \text{ m}\cdot\text{s}^{-2}) W=57.6 NW = 57.6 \text{ N}

    The magnitude of the weight is 57.6 N\boxed{57.6 \text{ N}}.

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