A diametrical tunnel is dug across the earth. A ball dropped into the tunnel from one side. The velocity of the ball when it reaches the centre of the earth is [Given: gravitational potential at the centre of earth `= -3//2(GM//R)`]

Explanation

As only the conservative gravitational force is acting,

`U_1 + K_1 = U_2 + K_2`
`U_1 = (-GMm)/( R) ,K_1 = 0`
`U_2 = (-3GMm)/(2R) , K_2 = 1/2 mv^2`
` rArr (-GMm)/(R ) + 0 = (-3)/(2) (GMm)/( R) + 1/2 mv^2`
`rArr (GMm)/(2R) = 1/2 mv^2 " or " v^2 = (GM)/( R)`
Now `mg = (GMm)/(R^2) " or " g = (GM)/(R^2) rArr v^2 = ((GM)/(R^2)) R = gR`
` rArr v = sqrt(gR)`