If the value of 'g' acceleration due to gravity, at earth surface is 10m /s² , its value in 2 m /s² at the centre of the earth, which is assumed to be a sphere of radius 'R' metre and uniform mass density is

The correct option is Zero.

To find the acceleration due to gravity at the center of the Earth, we can analyze how gravity varies with depth. The acceleration due to gravity $g\prime$ at a depth $d$ below the Earth's surface is given by the formula:
$g\prime =g\left(1-\frac{d}{R}\right)$
where:
• $g$ is the acceleration due to gravity at the surface of the Earth (10 m/s²),
• $R$ is the radius of the Earth,
• $d$ is the depth below the surface.

At the center of the Earth, the depth $d$ is equal to the Earth's radius $R$:
$d=R$

Substituting $d=R$ into our formula:
$g\prime =10\left(1-\frac{R}{R}\right)$
$g\prime =10\left(1-1\right)$
$g\prime =10\times 0=0\text{ m/s}{2}^{}$

Therefore, the acceleration due to gravity at the center of the Earth is zero because the gravitational forces exerted by the Earth's mass in all directions cancel each other out completely.