Mass M is divided into two parts xM and (1 – x)M. For a given separation, the value of x for which the gravitational attraction between the two pieces becomes maximum is

The correct option is 1/2.

Let us understand why this is the correct choice by breaking down the physical and mathematical concepts step-by-step.

Step 1: Formulating the Gravitational Force
According to Newton's Law of Gravitation, the gravitational force $F$ between two masses $m_1$ and $m_2$ separated by a distance $d$ is given by the formula:
$$F=\frac{G{m}_1{m}_2}{d^2}$$
where $G$ is the universal gravitational constant.

Here, the total mass $M$ is split into two pieces:
The first piece has mass $m_1=xM$.
The second piece has mass $m_2=(1-x)M$.

Substituting these mass values into Newton's gravitational force equation, we get:
$$F=\frac{G\left(xM\right)(1-x)M}{d^2}$$
Simplifying the expression by grouping the constants, we obtain:
$$F=\frac{G{M}^2}{d^2}·(x-x^2)$$

Step 2: Maximizing the Force using Calculus
For a given separation distance $d$ and a constant total mass $M$, the term $\frac{G{M}^2}{d^2}$ is a constant. Thus, the force $F$ depends solely on the variable function $f\left(x\right)=x-x^2$.

To find the value of $x$ that maximizes the gravitational attraction, we can take the first derivative of the force $F$ with respect to $x$ and set it equal to zero:
$$\frac{dF}{dx}=0$$
Differentiating our equation:
$$\frac{d}{dx}\left[\frac{G{M}^2}{d^2}(x-x^2)\right]=0$$
Since $\frac{G{M}^2}{d^2}\ne 0$, we focus on differentiating the terms inside the parentheses:
$$\frac{d}{dx}(x-x^2)=0$$
Applying basic derivative rules ($\frac{d}{dx}\left(x\right)=1$ and $\frac{d}{dx}\left(x^2\right)=2x$):
$$1-2x=0$$
Solving for $x$:
$$2x=1$$
$$x=\frac{1}{2}$$

Step 3: Verification of Maximality
To confirm that this value of $x$ yields a maximum, we can perform the second derivative test:
$$\frac{d^{2}F}{d{x}^2}=\frac{d}{dx}(1-2x)=-2$$
Because the second derivative is negative ($-2<0$), the point $x=\frac{1}{2}$ corresponds to a local maximum.

Therefore, the gravitational force of attraction between the two parts is maximized when the mass is divided exactly in half (i.e., $x=1/2$).