Questions: A planet has a mass of 2.50 × 10^23 kg and a surface gravity of 4.50 m / s^2. What is the radius of the planet? [?] × 10^[?] m

Solution Steps

We are given the mass of a planet and its surface gravity, and we need to find the radius of the planet. The mass of the planet is \(2.50 \times 10^{23} \, \text{kg}\) and the surface gravity is \(4.50 \, \text{m/s}^2\).

The formula for surface gravity \(g\) on a planet is given by:

\[ g = \frac{G \cdot M}{R^2} \]

where:

• \(g\) is the surface gravity,
• \(G\) is the gravitational constant, \(6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2\),
• \(M\) is the mass of the planet,
• \(R\) is the radius of the planet.

Rearrange the formula to solve for \(R\):

\[ R^2 = \frac{G \cdot M}{g} \]

\[ R = \sqrt{\frac{G \cdot M}{g}} \]

Substitute the known values into the formula:

\[ R = \sqrt{\frac{(6.674 \times 10^{-11} \, \text{m}^3/\text{kg/s}^2) \cdot (2.50 \times 10^{23} \, \text{kg})}{4.50 \, \text{m/s}^2}} \]

Calculate the value inside the square root:

\[ R = \sqrt{\frac{1.6685 \times 10^{13} \, \text{m}^3/\text{s}^2}{4.50 \, \text{m/s}^2}} \]

\[ R = \sqrt{3.7078 \times 10^{12} \, \text{m}^2} \]

\[ R \approx 1.9251 \times 10^{6} \, \text{m} \]

Final Answer