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An astronaut lands on a new, recently discovered planet in a different star system. The astronaut measures the acceleration due to gravity on the planet to be 12m/s2, and the mass of the planet is measured to be 7. 5E23kg. What is the radius of the new planet?

Sagot :

Supernova

Answer:

The radius of the new planet is ~2.04 * 10⁶ m, or 2,041,752 m.

Explanation:

We can use Newton's Law of Universal Gravitation:

  • [tex]\displaystyle F_g=G\frac{Mm}{r^2}[/tex]

Let's look at Newton's 2nd Law:

  • [tex]F=ma[/tex]

We can set these equations equal to each other:

  • [tex]\displaystyle G\frac{Mm}{r^2} =ma[/tex]

The mass of the second mass (astronaut) cancels out. We are left with:

  • [tex]\displaystyle G\frac{M}{r^2} =a[/tex]

We are solving for the radius of the new planet, so we can rearrange the equation:

  • [tex]\displaystyle r=\sqrt{\frac{GM}{a} }[/tex]

Substitute in our known values given in the problem (G = 6.67 * 10⁻¹¹ ; M = 7.5 * 10²³ ; a = 12).

  • [tex]\displaystyle r =\sqrt{\frac{(6.67\times 10^{-11})(7.5 \times 10^{23}}{12} }[/tex]
  • [tex]r=2.04 \times 10^6[/tex]

The radius of the new planet is ~2.04 * 10⁶ m.

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