A particle of mass m1 is kept at x = 0 and another of mass m2 at x = d. When a third particle is kept at x = d/4, it experiences no net gravitational force due to the two particles. Find m2/m1.
To solve this problem, you need to understand the principle of gravitational force and use Newton's law of gravitation.
First, let's consider the gravitational force between two particles. According to Newton's law of gravitation, the gravitational force between two particles is given by the formula:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the particles, and r is the distance between the particles.
In this problem, we want to find the ratio of m2 to m1. Let's assume that the third particle has a mass of m3.
Now, when the third particle is placed at x = d/4, it experiences no net gravitational force due to the two particles at x = 0 and x = d. This means that the gravitational forces acting on the third particle from the other two particles cancel each other out.
Let's calculate the net gravitational force acting on the third particle from the particle at x = 0:
F1 = G * (m1 * m3) / (d/4)^2
And let's calculate the net gravitational force acting on the third particle from the particle at x = d:
F2 = G * (m2 * m3) / (d/4)^2
Since the net gravitational force acting on the third particle is zero, F1 and F2 must be equal in magnitude and opposite in direction:
F1 = -F2
G * (m1 * m3) / (d/4)^2 = -G * (m2 * m3) / (d/4)^2
Simplifying the equation, we can cancel out the gravitational constant G and the term (d/4)^2:
m1 = -m2
Therefore, the ratio of m2 to m1 is -1.