Two bodies each of mass m are hung from a balance whose scale pans differ in a vertical height by h . If the mean density of the earth is , the error in weighing is

Step-by-step solution

Gravitational force on a body of mass m at height h due to earth, F= G M_e m (R+h)^2 where, M_e is mass of the earth. Density of earth, = mass of earth (M_e ) volume of earth (V) or M_e= V= ( 4 3 R^3 ) gathered F= G ( 4 3 R^3 ) m (R+h)^2 or F= G ( 4 3 R ) m (1+ h R )^2 or F= 4 3 G R m (1+ h R )⁻² gathered or F= 4 3 G R m (1+ h R )⁻² By using Binomial expansion, F= 4 3 G R m (1- 2 h R ) [ (1+x)^n=1+n x+ n(n-1) 2 ! x^2+ . and neglecting higher terms.] Difference in weight = 4 3 G R m- 4 3 G m R ( 2 h R ) Hence, error