QuestionJuly 5, 2025

What radius must the sphere have so that the acceleration due to gravity at the Earth's new surface would equal the acceleration due to gravity at the surface of the Sun? Express your answer using three significant figures and include the appropriate units. R=Value Units

What radius must the sphere have so that the acceleration due to gravity at the Earth's new surface would equal the acceleration due to gravity at the surface of the Sun? Express your answer using three significant figures and include the appropriate units. R=Value Units
What radius must the sphere have so that the acceleration due to gravity at the Earth's new surface would equal the acceleration due to gravity at the surface of the Sun? Express your answer using three significant figures and include the appropriate units. R=Value Units

Solution
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Answer

R = 637,000 \text{ m} Explanation 1. Identify the formula for gravitational acceleration Gravitational acceleration is given by g = \frac{GM}{R^2}, where G is the gravitational constant, M is the mass of the celestial body, and R is the radius. 2. Equate Earth's and Sun's gravitational accelerations Set \frac{GM_{\text{Earth}}}{R_{\text{new}}^2} = \frac{GM_{\text{Sun}}}{R_{\text{Sun}}^2} to find R_{\text{new}}. 3. Solve for R_{\text{new}} Rearrange to get R_{\text{new}} = R_{\text{Sun}} \sqrt{\frac{M_{\text{Earth}}}{M_{\text{Sun}}}}. 4. Substitute known values Use M_{\text{Earth}} = 5.972 \times 10^{24} \text{ kg}, M_{\text{Sun}} = 1.989 \times 10^{30} \text{ kg}, and R_{\text{Sun}} = 6.957 \times 10^8 \text{ m}.

Explanation

1. Identify the formula for gravitational acceleration<br /> Gravitational acceleration is given by $g = \frac{GM}{R^2}$, where $G$ is the gravitational constant, $M$ is the mass of the celestial body, and $R$ is the radius.<br />2. Equate Earth's and Sun's gravitational accelerations<br /> Set $\frac{GM_{\text{Earth}}}{R_{\text{new}}^2} = \frac{GM_{\text{Sun}}}{R_{\text{Sun}}^2}$ to find $R_{\text{new}}$.<br />3. Solve for $R_{\text{new}}$<br /> Rearrange to get $R_{\text{new}} = R_{\text{Sun}} \sqrt{\frac{M_{\text{Earth}}}{M_{\text{Sun}}}}$.<br />4. Substitute known values<br /> Use $M_{\text{Earth}} = 5.972 \times 10^{24} \text{ kg}$, $M_{\text{Sun}} = 1.989 \times 10^{30} \text{ kg}$, and $R_{\text{Sun}} = 6.957 \times 10^8 \text{ m}$.
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