Question

If the diameter of the earth becomes half of the present value but its average density remains unchanged then how would be the wieght of an object on earth been affected

Answer 1

There are two effects we need to consider, and they work in opposite directions.

The first is that if the radius of the earth were halved, then its volume would be divided by eight. Since its average density is the same, we can say its mass would be one eighth its current value. Thus, its gravitational field at any given position would be one eighth the strength.

However, the surface would also be twice as close to the earth’s centre. Gravitational field obeys an inverse square law: thus, the field at this distance would be four times stronger than the field where the earth’s surface used to be. This field—at the old surface radius—would be eight times weaker, so we know that the field at the earth’s new surface would actually be half its old value $(\because \frac{4}{8}=\frac{1}{2})$.

For this reason, objects at the surface would appear to have half the weight they do now.