Question

Two spheres of the same material and same radii r are touching each other. The gravitational force between the spheres is proportional to

• A. $\frac{1}{r}$
• B. $r^2$
• C. $\frac{1}{r^4}$
• D. $r^4$

Answer 1

Answer: (D)

$r^4$

Answer 2

Here's how to determine the proportionality:

1. Mass and Radius Relationship:
   Since the spheres are made of the same material, their density ($\rho$) is the same. The mass ($M$) of each sphere is proportional to its volume, which in turn is proportional to $r^3$.

   $$M \propto r^3$$

2. Distance Between Centers:
   Since the spheres are touching, the distance ($d$) between their centers is $2r$.

   $$d = 2r$$

3. Newton's Law of Gravitation:
   The gravitational force ($F$) between the spheres is given by Newton's Law of Gravitation:

   $$F = G \frac{M^2}{d^2}$$

   Where $G$ is the gravitational constant.

4. Substituting and Simplifying:
   Substitute $M \propto r^3$ and $d = 2r$ into the equation:

   $$F \propto \frac{(r^3)^2}{(2r)^2} = \frac{r^6}{4r^2} = \frac{1}{4}r^4$$

   Therefore, the gravitational force $F$ is proportional to $r^4$.

   $$F \propto r^4$$