Two particles, one heavy and the other light, placed at 50 cm from each other, are under the influence of gravitational force of one another. If mass of the heavier particle is 4 kg and its acceleration under the influence of gravitational force is $5 \times 10^{-10}\ ms^{-2}$, find the mass of the lighter particle.

Let the mass of the heavier particle be $M$ and the mass of the lighter particle be $m$.

Given:
$M = 4$ kg
Distance between the particles, $r = 50$ cm $= 0.5$ m
Acceleration of the heavier particle, $a_M = 5 \times 10^{-10}\ ms^{-2}$
Universal gravitational constant, $G \approx 6.674 \times 10^{-11}\ N\ m^2/kg^2$

The gravitational force exerted by the lighter particle on the heavier particle is:
$F = \dfrac{G M m}{r^2}$

According to Newton's second law, the force on the heavier particle is also given by:
$F = M a_M$

Equating the two expressions for force:
$M a_M = \dfrac{G M m}{r^2}$

$a_M = \dfrac{G m}{r^2}$

Rearranging to solve for the mass of the lighter particle $m$:
$m = \dfrac{a_M r^2}{G}$

Substituting the given values:
$m = \dfrac{5 \times 10^{-10} \times (0.5)^2}{6.674 \times 10^{-11}}$

$m = \dfrac{5 \times 10^{-10} \times 0.25}{6.674 \times 10^{-11}}$

$m = \dfrac{1.25 \times 10^{-10}}{6.674 \times 10^{-11}}$

$m = \dfrac{12.5}{6.674}$

$m \approx 1.8728$ kg

Answer: $1.8728$ kg