Question

A mass $M$ is split into two parts, $m_{0}$ and $M-m_{0}$. These two masses are then separated by a distance $D$. If the gravitational force between the parts is maximum, then the ratio $\frac{m_{0}}{M}$ is

• A. 0.2
• B. 0.4
• C. 0.5
• D. 0.6

Answer 1

Answer: (C)

According to given situation,

Again, $m_{0}$ and $M-m_{0}$ are kept at a distance $D$ gravitational force,

$F=\frac{G m_{0}\left(M-m_{0}\right)}{D^{2}}$
$F$ will be maximum, when
$\frac{d F}{d m_{0}}=0$
$\Rightarrow \frac{d}{d m_{0}} \frac{G m_{0}\left(M-m_{0}\right)}{D^{2}}=0$
$\Rightarrow \frac{d}{d m_{0}}\left(\frac{G m_{0} M-G m_{0}^{2}}{D^{2}}\right)=0$
$\Rightarrow \frac{G M-2 G m_{0}}{D^{2}}=0$
$\Rightarrow G M-2 G m_{0}=0$
$\Rightarrow M-2 m_{0}=0$
$\Rightarrow 2 m_{0}=M$
$\Rightarrow \frac{m_{0}}{M}=\frac{1}{2}=0.5$