Q4. The Earth’s mass is about 81 times the Moon’s. At what point on the line joining their centres is the net gravitational force on a spacecraft zero? Give its distance from the Earth’s centre as a fraction of the Earth–Moon distance d.
At distance x from the Earth, GMEm/x = GMMm/(d − x), so x/(d − x) = √81 = 9 and x = 0.9d. The point is nine-tenths of the way from the Earth to the Moon, considering only their instantaneous Newtonian pulls. It is not the Earth–Moon L1 point: remaining fixed relative to the orbiting bodies requires centripetal acceleration, not zero gravitational force.
Q5. Four equal masses m sit at the corners of a square of side a, and a mass M is at its centre. What is the net force on M? What is it if one of the corner masses is removed?
With all four, the pulls cancel in opposite pairs and the net force is zero. Removing one corner mass leaves the pull of the opposite corner uncancelled: the net force is GmM/s, towards that opposite corner, where s = a/√2 is the distance from the centre to a corner. That is 2GmM/a.
Q6. In NCERT Example 7.2, what is the force on the mass 2m at the centroid if the mass at A is made 3m instead of 2m? Take AG = 1 m.
The extra mass at A is 2m. The rest still cancels, so the net force is G × 2m × 2m/1 = 4Gm, towards A.