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 find the ratio m2/m1, we need to set up an equation that represents the gravitational forces acting on the third particle due to the two particles of mass m1 and m2.
Let's assume the third particle is of mass m3. The gravitational force on m3 due to m1 is given by Newton's law of gravitation as:
F1 = G * (m1 * m3) / (d/4)^2
where G is the gravitational constant.
Similarly, the gravitational force on m3 due to m2 is given by:
F2 = G * (m2 * m3) / (3d/4)^2
Since the net gravitational force on m3 is zero, we can equate these forces:
F1 + F2 = 0
Substituting the above equations and further simplifying, we get:
G * (m1 * m3) / (d/4)^2 + G * (m2 * m3) / (3d/4)^2 = 0
Simplifying further, we have:
m1 / (d/4)^2 + m2 / (3d/4)^2 = 0
m1 / (d^2 / 16) + m2 / (9d^2 / 16) = 0
Multiplying through by d^2/16, we get:
m1 + 9m2 = 0
Dividing both sides by m1:
1 + 9(m2/m1) = 0
9(m2/m1) = -1
m2/m1 = -1/9
Therefore, the ratio m2/m1 is -1/9.
To find the ratio of m2 to m1, we need to analyze the gravitational forces acting on the third particle when it is placed at x = d/4.
Let's denote the masses of the first and second particles as m1 and m2, respectively. Also, let the distance between them be d.
At x = d/4, the third particle is closer to the second particle than to the first particle. Therefore, we need to consider the gravitational forces due to both the first and second particles on the third particle.
The gravitational force F1 acting on the third particle due to the first particle is given by Newton's Law of Universal Gravitation:
F1 = G * (m1 * m3) / (r1)^2
where G is the gravitational constant, m3 is the mass of the third particle, and r1 is the distance between the first and third particles.
Similarly, the gravitational force F2 acting on the third particle due to the second particle is:
F2 = G * (m2 * m3) / (r2)^2
where r2 is the distance between the second and third particles.
Since the third particle experiences no net gravitational force, F1 and F2 must cancel each other out. Therefore, F1 = - F2 (the negative sign indicates opposite directions).
Let's substitute the values:
G * (m1 * m3) / (r1)^2 = - G * (m2 * m3) / (r2)^2
Since we are interested in the ratio of m2 to m1, we can cancel out the gravitational constant G and the mass of the third particle m3 from both sides of the equation:
(m1 / (r1)^2) = - (m2 / (r2)^2)
Now, we need to find the values of r1 and r2.
Given that the third particle is placed at x = d/4, the distance r1 between the first and third particles is:
r1 = (d/4)
And the distance r2 between the second and third particles is:
r2 = (3d/4)
Now, let's substitute the values of r1 and r2 back into the equation:
(m1 / (d/4)^2) = - (m2 / (3d/4)^2)
Simplifying:
(m1 / d^2/16) = - (m2 / 9d^2/16)
Multiplying both sides by (16d^2):
16(m1) = -9(m2)
Therefore, we find that m2/m1 = -16/9.
Note: The negative sign indicates that the force due to the second particle is opposite in direction to the force due to the first particle, ensuring no net gravitational force on the third particle.