I get it that there will be a gravitational force between objects attracted towards gravity but can there be a gravitational force between two objects resting on horizontal plane? In other words, does an object experience gravitational force in all directions?
Answer
The question is confusing, but I think you might possibly mean the following: Are two objects resting on a horizontal table gravitationally attracted to each other? If that's your question, then the answer is yes. The gravitational attraction is very weak for "normal-sized" objects, though. You can use the rule $$ F={Gm_1m_2\over r^2} $$ to work it out. In this formula, $G=6.67\times 10^{-11}\,{\rm N\,m^2/kg^2}$, $m_1,m_2$ are the masses, and $r$ is the separation between their centers. (Strictly speaking this is only correct if the objects are spheres.)
Cavendish managed to measure this attraction back in the 1700s, in a truly amazing experiment.
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