Question

The acceleration due to gravity is reduced by 50% at a height h above the surface of the earth and radius is halved then the new h in terms of R is?

1. R/8
2. 2R
3. R
4. 3R

Answer 1

Correct Answer - Option 1 : R/8

CONCEPT:

• Acceleration due to gravity: It is the acceleration gained by an object due to gravitational force.
  • Its SI unit is m/s². It has both magnitude and direction, hence, it’s a vector quantity.
  • Acceleration due to gravity is represented by g.
  • The standard value of g on the surface of the earth at sea level is 9.8 m/s²

Formula:

Acceleration due to gravity (g) = GM/R²

Acceleration due to gravity at height (g_H) =  g/(1 – 2h/R) 

where,G = universal gravitational constant (6.67×10-11 Nm2/kg2),g = acceleration due to gravity on the surface of the earth, M = mass of the earth,r = radius of the earth,h = height at which the body is

CALCULATIONS:

Given that, g₂= 75%, and g = 100, R₂ = 1/2 R

From the above formula,

(g_H) =  g/(1 – 2h/R) 

Putting all the above values in above and taking the ratio,

g₂ = g( 1 - 2h/R₂)

g₂/g = (1 - 2h/R₂)

50/100 = (1-4h/R).....(since R₂ = 1/2 R)

1/2 = R - 4h/R

R = 2R - 8h

8h = R

h = R/8

The correct option is 1.