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
$\frac{dF}{d{m}_0}=0$
$\Rightarrow \frac{d}{d{m}_0}\frac{G{m}_0\left(M-m_0\right)}{D^2}=0$
$\Rightarrow \frac{d}{d{m}_0}\left(\frac{G{m}_{0}M-G{m}_0^2}{D^2}\right)=0$
$\Rightarrow \frac{GM-2G{m}_0}{D^2}=0$
$\Rightarrow GM-2G{m}_0=0$
$\Rightarrow M-2m_0=0$
$\Rightarrow 2m_0=M$
$\Rightarrow \frac{m_0}{M}=\frac{1}{2}=0.5$