Physics

Question

The radius of the planet saturn is 6.03 × 107 m, and its mass is 5.68 × 1026 kg.
a. find the density of saturn (its mass divided by its volume) in grams per cubic centimeter. (the volume of a sphere is given by πr 3.)4 __3
b. find the surface area of saturn in square meters. (the surface area of a sphere is given by 4πr 2 .)

2 Answer


  • You wouldn't know it from reading the question,
    but the volume of a sphere is
                                                           4/3 π (radius)³ .

    So the volume of Saturn is

             (4/3 π) (6.03 x 10⁷ m)³  =  about     9.184 x 10²³ m³

    and its density is      (mass) / (volume)

               = (5.68 x 10²⁶ kg) / (9.184 x 10²³ m³)

    Sadly, we need this to be in units of  ' gram/cm³ ' so we need to
    account for some conversion of units.

          = (5.68 x 10²⁶ kg) · (1,000 g/kg) / (9.184 x 10²³ m³) · (10⁶ cm³/m³)

         =  (5.68 x 10²⁹ grams)  /  (9.184 x 10²⁹ cm³)

         =       0.618  gram/cm³  .

    Look at that !
    The density of the planet Saturn is less than  ' 1 '.
    If you had a big enough bathtub full of water, Saturn would float in it ! 


    The surface area of a sphere is

                                           4  π  (radius)²

                                      =  (4 π) (6.03 x 10⁷ m)²

                                      =  (4 π) (36.4 x 10¹⁴ m²)

                                      =     4.57 x 10¹⁶ m²      

              



  • a) The density of the planet Saturn is [tex]\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}[/tex].

    b) The surface area of the planet Saturn is [tex]\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}[/tex].

    Further Explanation:

    Given:

    The radius of the planet Saturn is [tex]6.03 \times {10^7}\,{\text{m}}[/tex].

    The mass of the planet Saturn is [tex]5.68 \times {10^{26}}\,{\text{kg}}[/tex].

    Concept:

    The planet is considered to be a huge spherical body. Therefore, the volume of the planet is expressed as:

    [tex]V = \frac{4}{3}\pi {r^3}[/tex]

    Substitute the value of [tex]r[/tex] in the above expression.

    [tex]\begin{aligned}V&=\frac{4}{3}\pi\times{\left({6.03\times{{10}^9}\,{\text{cm}}}\right)^3}\\&=4.189\times2.19\times{10^{29}}\\&=9.17\times{10^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}\\\end{aligned}[/tex]

    Part (a):

    The density of the planet can be expressed as:

    [tex]\rho  = \frac{{{\text{mass}}}}{{{\text{Volume}}}}[/tex]

    Substitute the values of mass and volume of the planet.

    [tex]\begin{aligned}\rho&=\frac{{5.68\times{{10}^{26}}\,{\text{kg}}}}{{9.17 \times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=\frac{{5.68\times{{10}^{29}}\,{\text{g}}}}{{9.17\times{{10}^{29}}\,{\text{c}}{{\text{m}}^{\text{3}}}}}\\&=0.619\,{{\text{g}}\mathord{\left/{\vphantom{{\text{g}}{{\text{c}}{{\text{m}}^{\text{3}}}}}}\right. \kern-\nulldelimiterspace}{{\text{c}}{{\text{m}}^{\text{3}}}}}\\\end{aligned}[/tex]

    Thus, the density of the planet Saturn is [tex]\boxed{0.619\,{{\text{g}} \mathord{\left/ {\vphantom {{\text{g}} {{\text{c}}{{\text{m}}^{\text{3}}}}}} \right. \kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^{\text{3}}}}}}[/tex].

    Part (b):

    The total surface area of the spherical planet is given as:

    [tex]A = 4\pi {r^2}[/tex]

    Substitute the value of [tex]r[/tex] in the above expression.

    [tex]\begin{aligned}A&=4\pi\times{\left({6.03\times{{10}^7}\,{\text{m}}}\right)^2}\\&= 12.56 \times3.63\times{10^{15}}\\&=4.569\times{10^{16}}\,{{\text{m}}^{\text{2}}} \\ \end{aligned}[/tex]

    Thus, the surface area of the planet Saturn is [tex]\boxed{4.57 \times {{10}^{16}}\,{{\text{m}}^{\text{2}}}}[/tex].

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    Answer Details:

    Grade: High School

    Subject: Physics

    Chapter: Units and Measurement

    Keywords:

    Radius of the planet, Saturn, mass of Saturn, density, grams per cubic centimeter, 5.68x10^26 kg, 6.03x10^7 m, volume of sphere.

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