three bodies of masses m,2m wnd 3m are placed at point A,B and X respectively of an equilateral triangle..Calculate the magnitude of resuptant gravitational force of an body B due to the bodies at A and B
Answers
Explanation:
Draw a perpendicular AD to the side BC.
∴AD=ABsin60=
2
3
l
Distance AO of the centroid O from A is
3
2
AD.
∴AO=
3
2
(
2
3
l)=
3
l
By symmetry, AO=BO=CO=
3
l
Force on mass 2m at O due to mass m at A is
F
OA
=
(l/
3
)
2
Gm(2m)
=
l
2
6Gm
2
along OA
Force on mass 2m at O due to mass m at B is
F
OB
=
(l/
3
)
2
Gm(2m)
=
l
2
6Gm
2
along OB
Force on mass 2m at O due to mass m at C is
F
OC
=
(l/
3
)
2
Gm(2m)
=
l
2
6Gm
2
along OC
Draw a line PQ parallel to BC passing through O. Then ∠BOP=30=∠COQ
Resolving
F
ˉ
OB
and
F
ˉ
OC
into two components.
Components acting along OP and OQ are equal in magnitude and opposite in direction. So, they will cancel out while the components acting along OD will add up.
∴ The resultant force on the mass 2m at O is
F
R
=F
OA
−(F
OB
sin30+F
OC
sin30)
=
l
2
6Gm
2
−(
l
2
6Gm
2
×
2
1
+
l
2
6Gm
2
×
2
1
)=0