Three particles, two with masses m and one with mass M, might be arranged in any of the four configurations shown below. Rank the configurations according to the magnitude of the gravitational force on M, least to greatest
For configuration (i), gravitational force on M due to m and m
F1=
GMm
d2
+
GMm
(2d)2
=
GMm
d2
[1+
1
4
] =
GMm
d2
.
5
4
........(i) For configuration (ii), gravitational force on M due to m and m
F2=
GMm
d2
−
GMm
d2
=0 ........(ii) For configuration (iii), gravitational force on M
F3=√(F′)2+(F")2 (∵ angle between F′ and F" is 90° ) =√(
GMm
d2
)2+(
GMm
d2
)2=
GMm
d2
√2 .........(iii) For configuration (iv), gravitational force on M, where 0<θ<90°
F4=√(F′)2+(F")2+2F′F"‌cos‌θ =√(
GMm
d2
)2+(
GMm
d2
)2+2
GMm
d2
.
GMm
d2
‌cos‌θ =
GMm
d2
√1+1+2‌cos‌θ =
GMm
d2
√2(1+cos‌θ) =(
GMm
d2
√2)√1+cos‌θ ..........(iv) (∵ for 0<θ<90°,√1+cos‌θ>) From Eqs. (i), (ii), (iii) and (iv), we get F2<F1<F3<F4 Thus, the correct option is (b).