The free-fall acceleration on the surface of Jupiter is about two and one half times that on the surface of the Earth. The radius of Jupiter is about 11.0 RE (RE = Earth's radius = 6.4 106 m). Find the ratio of their average densities, ρJupiter/ρEarth.
5/22
AKBAR
To find the ratio of the average densities of Jupiter (ρJupiter) and Earth (ρEarth), we need to make use of the given information about the free-fall acceleration and the radius of Jupiter.
First, let's determine the free-fall acceleration on the surface of Jupiter. The free-fall acceleration, denoted as g, is given by the equation:
g = GM / R²,
where G is the gravitational constant, M is the mass of the celestial body, and R is the radius of the celestial body.
Given that the free-fall acceleration on the surface of Jupiter (gJupiter) is two and a half times that on the surface of Earth (gEarth), we can set up the following equation:
gJupiter = 2.5 * gEarth.
Next, we'll use the fact that g = GM / R² to relate the free-fall accelerations to the radii of the celestial bodies.
For Earth:
gEarth = GMearth / RE²,
where MEarth is the mass of Earth.
Similarly, for Jupiter:
gJupiter = GMjupiter / RJupiter²,
where MJupiter is the mass of Jupiter.
We're given the ratio of their radii as:
RJupiter / RE = 11.0.
Now, let's substitute the expressions for gEarth and gJupiter into the equation gJupiter = 2.5 * gEarth:
GMjupiter / RJupiter² = 2.5 * (GMearth / RE²).
Next, we can rearrange the equation to find the ratio of the average densities of Jupiter and Earth:
ρJupiter / ρEarth = (Mjupiter / Vjupiter) / (Mearth / Vearth),
where Mjupiter and Mearth are the masses of Jupiter and Earth, respectively, and Vjupiter and Vearth are their respective volumes.
Using the equations for volume:
Vearth = (4/3) * π * RE³,
Vjupiter = (4/3) * π * RJupiter³,
we can substitute these expressions into the density ratio equation:
ρJupiter / ρEarth = (Mjupiter / [(4/3) * π * RJupiter³]) / (Mearth / [(4/3) * π * RE³]).
Finally, we can simplify the equation by canceling out the common factors:
ρJupiter / ρEarth = (Mjupiter / RJupiter³) / (Mearth / RE³) * (RE³ / RJupiter³).
Substituting the given ratio of their radii (RJupiter / RE = 11.0):
ρJupiter / ρEarth = (Mjupiter / RJupiter³) / (Mearth / RE³) * (RE³ / (11.0 * RE)³).
Now, you can calculate the ratio of their average densities by plugging in the values for the mass of Jupiter and Earth and solving the equation.
Note: The mass of Jupiter (Mjupiter) is approximately 1.898 × 10^27 kg, and the mass of Earth (Mearth) is approximately 5.972 × 10^24 kg.