How does the weight of an object vary with respect to mass and radius of the earth. In a hypothetical case, if the diameter of the earth becomes half of its present value and its ,mass becomes four times of its present value, then how would the weight of any object on the surface of the earth be affected?

Explanation

Let radius and mass of the earth be R,M respectively and mass of object be m. Weight of object ,
`w=mg=(GMm)/(R^(2))" "...(i)[becauseg=(Gm)/(R^(2))]`
According to the question, the diameter of the earth becomes half of its present value. So, radius also becomes half of its initial values
`So," "R^(')=(R)/(2)`
Again, mass of earth becomes four times of its present value.
`So," "M^(')=4M`
Now, weight of the object
`w^(')=(GM^(')m)/(R^('))`
Put `R^(')=(R)/(2) and M^(')=4m` in above formula, we get
`w^(')=(G4Mm)/((R//2)^(2))`
`=(4GMm)/(R^(2)/4)`
`=(16GMm)/(R^(2))`
`"From Eq.(i), we get"" " w^(')=16w`
Thus, weight of the object will become 16 times of its original weight.