The weight of an object at the centre of the earth of radius $R$ is

What is the effect on the weight of a body due to the revolution of the earth?

$(a)$ The weight of a man on the surface of the earth is $392 \, N$. Find his mass. $(g = 9.8 \, m s^{-2})$
$(b)$ If the man were taken to the moon,what would be:
$(i)$ his mass and
$(ii)$ his weight?
$(iii)$ Calculate the acceleration due to gravity on the moon.

$(a)$ The Earth is acted upon by the gravitation of the Sun,yet it does not fall into the Sun. Why?
$(b)$ An object of mass $500 \, g$ is dropped from a tower of height $5 \, m$. Calculate its momentum when it touches the ground (given $g = 10 \, m s^{-2}$).

$(a)$ $A$ cube of side $5\, cm$ is immersed in water and then in a saturated salt solution. In which case will it experience a greater buoyant force? If each side of the cube is reduced to $4\, cm$ and then immersed in water,what will be the effect on the buoyant force experienced by the cube as compared to the first case for water? Give reasons for each case.
$(b)$ $A$ ball weighing $4\, kg$ with a density of $4000\, kg\, m^{-3}$ is completely immersed in water of density $10^3\, kg\, m^{-3}$. Find the force of buoyancy on it. (Given $g = 10\, m\, s^{-2}$)

State whether the following statement is True or False:
The acceleration of a body thrown vertically upward is numerically the same as the acceleration of a body falling downward,but opposite in sign.