Question

An object of mass 10 kg is divided into two parts,
which are placed at a distance d. If the gravitational
force of attraction is maximum between the parts, then
the ratio of the mass of both parts is



answer fast pls​

Answer 1

Given:

Mass of the object = 10 kg

When the object is divided into parts they are kept at a distance = d

To find:

The ratio of mass of both the parts.

Solution:

Let one part's mass be x then the other part's mass will be 10-x.

The gravitational force of attraction between them will be: F = G(x)(10-x)/ d²

If the force is maximum then x(10-x) has to be maximum because all others are constants.

To find the max value of 10x- x^2, differentiate the equation and put that equal to 0.

10 - 2x= 0

2x = 10

x = 5

Again differentiating the equation w.r.t x we can determine whether the value is minimum or maximum.

So differentiating 10 - 2x=0

We get -2, so the double differentiation of equation is negative hence x=5 will give the maxima of the equation.

Therefore 10x-x² will be max at x=5

So the mass of the two parts will be 5 kg each.

Ratio of their mass = 5/5 = 1:1

Therefore the required ratio is 1:1.

Answer 2

Given:

An object of mass 10 kg is divided into two parts,

which are placed at a distance d.

To find:

Ratio of both parts of gravitation force is max?

Calculation:

Let one part be m and the other part be (10-m):

So, gravitational force:

$\therefore \: F = G \dfrac{m(10 - m)}{ {d}^{2} }$

$\implies\: F = G \: \dfrac{10m - {m}^{2} }{ {d}^{2} }$

$\implies\: F = \dfrac{G}{ {d}^{2} } \bigg(10m - {m}^{2} \bigg)$

Now, for maxima, dF/dm = 0 and d²F/dm² < 0:

$\implies\: \dfrac{dF}{dm} = \dfrac{d \bigg \{\dfrac{G}{ {d}^{2} } \bigg(10m - {m}^{2} \bigg) \bigg \}}{dm}$

$\implies\: \dfrac{dF}{dm} = \dfrac{G}{ {d}^{2} } \bigg(10 - 2m \bigg)$

$\implies\: \dfrac{G}{ {d}^{2} } \bigg(10 - 2m \bigg) = 0$

$\implies\: 10 - 2m= 0$

$\implies \: m = 5 \: kg$

Now, 2nd order differentiation:

$\implies\: \dfrac{ {d}^{2} F}{d {m}^{2} } = \dfrac{d \bigg \{\dfrac{G}{ {d}^{2} } \bigg(10 - 2m \bigg) \bigg \}}{dm}$

$\implies\: \dfrac{ {d}^{2} F}{d {m}^{2} } = \dfrac{G}{ {d}^{2} } ( - 2)$

$\implies\: \dfrac{ {d}^{2} F}{d {m}^{2} } < 0$

Hence maxima when m = 5.

• So, one mass (m) = 5 kg
• Other mass (10-m) = (10-5) = 5 kg

So, ratio is 1 : 1.