Two bodies A and B having masses 2kg and 4kg respectively are separated by 2m. Where should a body of mass 1kg be placed so that the gravitational force on this body due to bodies A and B is zero?
Answers
ANS:
Given data-
1. Mass A = 2Kg
2) Mass B = 4Kg
3) Mass C= 1kg
Distance between the bodies = 2mtr
Now we are required to find a point between these 2 masses where if a small mass of 1Kg is put then NET gravitational force on it exerted by remaining other 2 masses will be zero
Formula of gravitational force between 2 objects is given by:
G M₁ M₂ / [distance between them]²
where,
G= Gravitational constant
M₁ and M₂ are masses
now assume mass c is put somewhere between A ans B such that net gravitational force is zero and also distance of C from A is X meter
therefore remaining distance of C from B will be (2-X)
In Other words ,
force between A-C = force between C-B
G M(A) * M(C)/ X² = G M(C) * M(B) / [2-X]²
after cancelling equal terms we will be left with this:
M(A) /M(B) = X²/[2-X]²
2/4 = X²/[2-X]² => 1/2 = X²/[2-X]²
taking both sides under root
1/√2 = X/ (2-X) => 2-X = √2 X
2 = √2 X- X or X(2-√2) = 2
X = (1 - 1/√2) final ans
Answer:
Here, mass of body A,m
1
=2kg, mass of body B,m
2
=4kg
distance between the bodies A and B=2m
mass of body C,m=1kg
Let body C be placed at a distance x from A,
Thus, AC=x and BC=(2−x)
As gravitational force on C due to A= gravitational force on C due to B,
F
1
=F
2
or
x
2
Gm
1
m
=
(2−x)
2
Gm
2
m
or
x
2
m
1
=
(2−x)
2
m
2
or
x
2
2
=
(2−x)
2
4
or2x
2
=(2−x)
2
Taking square root of both sides, we get
2x
=2−x
or 1.414x+x=2orx=
2.414
2
=0.83m