A mass M is split into two parts, m and M – m, which are then separated by a certain distance. What ratio of m/M maximizes the gravitational force between the two parts

The correct option is 1/2.

To find the ratio of $m/M$ that maximizes the gravitational force between the two parts, we start by expressing the gravitational force using Newton's Law of Universal Gravitation.

Let the total mass be $M$. It is split into two parts: one part of mass $m$ and the other part of mass $M-m$. Let $r$ be the distance of separation between them. The gravitational force $F$ between the two masses is given by:
$F=\frac{Gm(M-m)}{r^2}$
where $G$ is the universal gravitational constant.

To find the value of $m$ that maximizes the force $F$, we differentiate $F$ with respect to $m$ and set the derivative to zero:
$\frac{dF}{dm}=0$

Since $G$, $M$, and $r$ are constants, we only need to differentiate the term $m(M-m)$:
$\frac{d}{dm}(mM-m^2)=0$

Differentiating each term with respect to $m$ gives:
$M-2m=0$

Solving for $m$:
$M=2m$

Rearranging this to find the ratio $m/M$:
$\frac{m}{M}=\frac{1}{2}$

Thus, the ratio of $m/M$ that maximizes the gravitational force between the two parts is 1/2.