If g₁ is the gravitational acceleration of the earth at a height equal to half of the radius of the earth and g₂ is the gravitational acceleration of the earth at a depth equal to half of the radius of the earth, then find the ratio of the g₁ to g₂.

CONCEPT:

Acceleration Due to Gravity:

• The force of attraction exerted by the earth on a body is called gravitational pull or gravity.
• We know that when a force acts on a body, it produces acceleration. Therefore, a body under the effect of gravitational pull must accelerate.
• The acceleration produced in the motion of a body under the effect of gravity is called acceleration due to gravity, it is denoted by g.

• The acceleration due to gravity on the surface of the earth is given by,

• The acceleration due to gravity above the surface of the earth is given as,

If h <<< R

• The acceleration due to gravity below the surface of the earth is given as,

CALCULATION:

• The gravitational acceleration of the earth at a height equal to half of the radius of the earth is given as (h = R/2),

     -----(1)

• The gravitational acceleration of the earth at a depth equal to half of the radius of the earth is given as (d = R/2),

     -----(2)

By equation 1 and equation 2,

• Hence, option 2 is correct.

Additional Information

• The acceleration due to gravity below the surface of the earth is given as,
• Let M be the mass of the earth, R be the radius of the earth g_d be gravitational acceleration at depth d from the earth surface
• g be gravitational acceleration on the earth's surfaces.
• ρ  be the density of the earth.
• P be the point inside the earth at depth d from earth surfaces.

∴ CS - CP = d, ∴ CP = R - d      ..............(1) (since CS = R)

∴ g = 

∴  .............(2)

gd = acceleration due to gravity at depth 'd'

∴ 

∴  ..........(3)

Dividing eq. (3) by eq (2)

∴