Answer
The mass of particle C is $~~1.0~kg$
Work Step by Step
$F_{net,x} = 0$ when $x = 0.4~m$
Then the gravitational force due to particle C must be equal in magnitude to the gravitational force due to particle A when particle C is at position $x = 0.4~m$
In part (a), we noted that the magnitude of the gravitational force due to particle A is $4.17\times 10^{-10}~N$
We can find the mass of particle C:
$F = \frac{G~m_C~m_B}{(0.40~m)^2} = 4.17\times 10^{-10}~N$
$m_C = \frac{(4.17\times 10^{-10}~N)(0.40~m)^2}{G~m_B}$
$m_C = \frac{(4.17\times 10^{-10}~N)(0.40~m)^2}{(6.67\times 10^{-11}~N~m^2/kg^2)(1.0~kg)}$
$m_C = 1.0~kg$
The mass of particle C is $~~1.0~kg$
