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

Options

A. `(4pi rho Gmh)/(3)`
B. `(3pi rho Gmh)/(4)`
C. `(8pi rho Gmh)/(3)`
D. `(3pi rho Gmh)/(8)`

Answer

C

Explanation

Gravitational force on a body of mass m at height h due to earth,
` F = (GM_(e) m)/((R + h)^(2))`
where, `M_(e)` is mass of the earth.
`therefore` Density of earth, `rho = ("mass of earth"(M_(e )))/("volume of earth" (v))`
or ` M_(e ) = rho. v = rho ((4)/(3) pi R^(3) ) `
`therefore " " F = (G ((4)/(3) pi R^(3))rho m)/((R + h)^(2))`
or `" " F = (G ((4)/(3) pi R^(3))rho m)/((1 + (R)/(h))^(2))`
or `" " F = (4)/(3) pi GR rho m (1 + (h)/(R) )^(-2)`
By using Binomial expansion,
` F = (4)/(3) pi Gr rho m (1 - (2h)/(R ) )`
` [ because ( 1 + x)^(n) = 1 + nx + (n (n - 1))/(2l) x^(2) + ..... and "neglecting higher terms". ]`
Difference in weight = `(4)/(3) pi GR rho m - (4)/(3) pi G rho mR ((2h)/(R ) )`
Hence, error in weight =` (8)/(3) pi rho ` Gmh.