The weight of the sphere can be determined using Archimedes` principle, which states that the buoyant force acting on an object immersed in a fluid is equal to the weight of the fluid displaced.

In this scenario, the lower half of the sphere is immersed in the liquid of density p2, while the upper half is immersed in the oil of density pj.

First, let`s analyze each option provided:

Option 1: V(p2 - pj)/2

This option suggests that the weight of the sphere is equal to half of the difference in densities between the liquid and the oil. However, this does not take into account the gravitational force acting on the fluid.

Option 2: V(p2 + pj)g/2

This is the correct option. It considers both densities and incorporates the acceleration due to gravity, represented by g. By adding the densities of p2 and pj and dividing by 2, we obtain the average density of the sphere. Multiplying this by the volume V and the acceleration due to gravity g gives us the weight of the sphere.

Option 3: V(p2 + pj)

This option is the sum of the densities without considering the gravitational force. It disregards the weight of the fluid displaced by the sphere.

Option