An object of mass M is divided into two parts ,which are placed at distance d.if gravitational force of attraction is maximum between parts,then find the ratio of mass of both parts
If the parts are x and M-x, then
F = Gx(M-x)/r^2
= G/r^2 (Mx-x^2)
This is just a parabola, with vertex at M/2. That is, the force is maximum when the masses are equal.
It's like the area of a rectangle with constant perimeter is greatest if it is a square.
Why we differentiate wrt m?
To determine the ratio of mass between the two parts when the gravitational force of attraction is maximum, we need to consider the concept of center of mass.
The center of mass of an object is the average position of all the parts of the object, taking into account their respective masses. In this case, if the gravitational force of attraction is maximum, it means the two parts are placed such that their centers of masses lie at the same position.
Let's assume that the mass of one part is m1 and the other part is m2. The total mass of the object is M, which is the sum of m1 and m2 (M = m1 + m2).
Given that the distance between the two parts is d, the center of mass will be located at the midpoint between them. Therefore, the position of the center of mass will be d/2 from each part.
To find the center of mass position, we use the equation:
(center of mass) = (m1 * position of m1 + m2 * position of m2) / (m1 + m2)
Substituting the values, we get:
d/2 = (m1 * 0 + m2 * d) / (m1 + m2)
Simplifying, we have:
d/2 = m2d / (m1 + m2)
Cross-multiplying, we get:
dm1 + dm2 = 2m2d
Rearranging the terms, we get:
2m2d - dm2 = dm1
Dividing throughout by m2, we get:
2d - d(m2/m1) = m1
Finally, dividing both sides by m1, we obtain:
2d/m1 - d(m2/m1^2) = 1
We can express the ratio of the masses as m2/m1:
m2/m1 = (2d/m1 - 1) / d
Hence, the ratio of the masses of the two parts is (2d/m1 - 1) / d.