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Question

Three particles, each of mass m, are placed at the vertices of an equilateral triangle of side a. The net gravitational force on a body of mass M at the centroid of the triangle is


A

zero

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B

GmMa2

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C

2GmMa2

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D

3GmMa2

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Solution

The correct option is A

zero


Given that AB = BC = CA = a. The perpendiculars from A, B and C on opposite sides meet at the centroid O and bisect the sides AB, BC and CA. Let AO = BO = CO = r. Centroid also divides the lines AD, BE and CF in the ratio 2 : 1, i.e. AO=23AD, BO=23BE and CO=23CF.

In triangle ABD, AD=a sin 60=3a2
Similarly, BE = CF =3a2
r=AO=BO=CO=23×3a2=a3
The net force exerted on a unit mass placed at O due to three equal masses m at vertices A, B and C.

Since the three masses are equal and their distances from O are also equal, they exert forces FA, FB and FC of equal magnitude. Their directions are shown in the figure. It follows from the symmetry of forces that their resultant at point O is zero.


Hence, the correct choice is (a).


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