Question

what happens to the potential energy if the mass of object is doubled or halved

Answer 1

P.E. = m*v

If mass is doubled, PE also gets doubled.
If mass is halved, PE also gets halved.

Answer 2

Potential Energy = P.E = mgh
m = mass 
h = height
g = acceleration due to gravity

P.E = mgh
New potential energy = P.E '
When mass is doubled , 

P.E ' = (2m)gh

P.E ' = 2 × mgh
P.E '  = 2 × P.E

Therefore , 
When mass is doubled ,  Potential energy gets doubled.

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P.E = mgh
New potential energy = P.E '

When mass is halved :

P.E ' = (1/2 m)g h
P.E ' = 1/2 ×mgh
P.E' = 1/2 × P.E

When mass is halved ,  Potential energy decreases by half.