Question

'if diametre of earth is reduced to half mass remains constant what is then duration of day and night'

Answer 1

angular momentum remains the same.

I2 omega 2 = I1 omega 1

I = constant * m R^2

If m2 = m1
and
R2 = .5 R1
then
m R2^2 = .25 m R1^2
and I2 = (1/4) I1
(1/4)I1 omega2 = I1 omega1
omega2 = 4 omega1
T2 = (1/4) T1
day is 6 hours long instead of 24 so at the equator at equinox we have 3 hours dark and three hours light. :)

Answer 2

Come to think of it, anywhere at equinox you will have three hours light and three hours dark