Gravitational doubt If we can shield a charge from electrical forces by putting it inside a hollow conductor. Can we shield a body from gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means ?
 A: Shielding works because the electrons (or other free charge particles) migrate in such a way that the electric field the conductor generates inside itself is equal and opposite of the external field applied to the outside.
This only works when there is a repulsion force.
A: No. Shielding works in electrostatics because there are both positive and negative electric charges. In gravitation, there is only one kind of charge, i.e. mass, which is positive. As a consequence, gravity can only attract, and there is no way to counteract this by a repulsive force also originating from gravitational effects.  
A: Your question brings up an interesting sidelight.  Schiff and Barnhill[1] showed that an electron inside a conducting cylinder will be subjected to an induced electric field that will counteract the pull of gravity.  Their reasoning can be summarized as follows.  


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*The atoms in the crystalline metal that the cylinder is made up of do not fall through each other because of the Coulomb forces between the electrons of the individual atoms. 

*When the conducting cylinder is placed with it's axis of rotation perpendicular to the ground, the force of gravity places an additional force on each atom that makes up the cylinder.

*The atoms still do not fall through one another and Barnhill and Schiff surmise this is due to an increase in the electric field between the electrons surrounding each atom.  The increase in field is just enough to support each neighboring electron against the downward force of gravity.

*Near the surface of the cylinder, this electric field that separates the electrons in neighboring atoms can be observed outside the cylinder wall.

*If the cylinder is long enough and narrow enough,the electric field inside the cylinder will be uniform.  An electron introduced inside the cylinder will simply float in this induced field which is strong enough to support the weight of an electron against gravity.
One of the reasons  physicists care about all of this is that we're interested in finding out if the anti-particle of the electron, (the positron) falls up or down in gravity[3].  To test this, we need to understand what the positron will do if we contain it in a metal vessel, a cylinder for example, to perform the experiments.  If positrons fall down under gravity and Schiff and Barnhills' theory was correct, then a positron should fall down through the cylinder at twice it's normal rate of descent.  If they fall up under a gravitational pull, then the positron should also be suspended within the cylinder.
This might all sound a bit outlandish, but Witteborn, and Fairbank performed an experiment that confirmed Schiff and Barnhill's theory.  There were some issues with the experiment.  The experimenters surmised that the walls between crystaline domains within the material making up the cylinder should have influenced the results.  It appears that they didn't.  This behavior is not  yet theoretically understood and consequently the results of the experiment are not widely agreed upon.
References


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*Schiff and Barnhill
https://journals.aps.org/pr/abstract/10.1103/PhysRev.151.1067

*Witteborn and Fairbank
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1049

*More on falling positrons
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.64.237
