A small ball of mass 'm' is released at a height 'R' above the earth surface, as shown in the figure. If the maximum depth of the ball to which it goes is R/2 inside the earth through a narrow grove before coming to rest momentarily. the grove, contain an ideal spring of spring constant K and natural length R, the value of K is (R is radius of Earth and M mass of Earth)

Options

A. `(3GMm)/(R^(3))`
B. `(6GMm)/(R^(3))`
C. `(9GMm)/(R^(3))`
D. `(7GMm)/(R^(3))`

Answer

D

Explanation

By energy conservation
`K_(i)+U_(i)=K_(f)+U_(f)`
`0-(GMm)/(2R)=(1)/(2)K((R)/(2))^(2)(11GMm)/(8R)`
`(GMm)/(R)[(11)/(8)-(1)/(2)]=(1)/(2)K((R^(2))/(4))`
`(7GMm)/(8R)=(KR^(2))/(8)`
`K=(7GMm)/(R^(3))`