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As you know that
F=G R^2/Mm
So at centre of the earth, R=0
So, F=0
And weight is force exerted,
Therefore, Weight is zero at the centre of the earth.
Hope it helps :)
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when we are at the center of the earth our weight becomes 0 because at the center acceleration due to gravity is 0 and acceleration due to gravity is 0 because their is no radius in the center of earth.and since weight =m*g and since g is 0 therefore F=m*0 ,F=0. because of this our weight becomes 0 at the center of the earth.
   i hope that this will help u and if it helps u plzzzz...mark it as the brainiest. 
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