A spherical planet has a mass $M_p$ and diameter $D_p$. A particle of mass $m$ falling freely near the surface

Question

A spherical planet has a mass $M_p$ and diameter $D_p$. A particle of mass $m$ falling freely near the surface of this planet will experience acceleration due to gravity equal to:

Accepted Answer

1. **Formula for Acceleration Due to Gravity:** The acceleration due to gravity ($g$) on the surface of a planet with mass $M_p$ and radius $R_p$ is derived from Newton's Law of Universal Gravitation. The gravitational force on a mass $m$ is $F = \frac{GM_p m}{R_p^2}$. By Newton's second law, force $F = mg$. Therefore, the acceleration is $g = \frac{F}{m} = \frac{GM_p}{R_p^2}$ [Equation 7.9].
2. **Substitute Diameter:** The problem provides the diameter $D_p$. The radius is half the diameter: $R_p = \frac{D_p}{2}$.
3. **Calculation:** Substitute $R_p$ into the equation for $g$:
   $$ g = \frac{GM_p}{(D_p/2)^2} = \frac{GM_p}{D_p^2 / 4} = \frac{4GM_p}{D_p^2} $$
   Note that acceleration due to gravity is independent of the mass of the falling particle ($m$).

Step-by-Step Solution

Practice All PHYSICS Questions