$A$ body of mass $10 \,kg$ is taken to the centre of the earth. What will be its mass there (in $,kg$)?

$A$ stone dropped from a certain height takes $4$ seconds to reach the ground. The height is: (in $, m$)

$(a)$ Prove that if the Earth attracts two bodies placed at the same distance from the center of the Earth with equal force,then their masses will be the same.
$(b)$ Mathematically express the acceleration due to gravity experienced by a free-falling object.
$(c)$ Why is $G$ called a universal gravitational constant?

How does the weight of an object vary with respect to the 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 its present value,then how would the weight of any object on the surface of the Earth be affected?

Write three differences between gravitational acceleration $(g)$ and gravitational constant $(G).$

$(i)$ Is the acceleration due to gravity of earth '$g$' always a constant at every place? Discuss.
$(ii)$ During a free fall,will heavier objects accelerate more than lighter ones? Give reason for your answer.