The center of gravity of an object is at the same position as the center of mass

Question

The center of gravity of an object is at the same position as the center of mass when

a. the object is so far from any mass that .
b.the object is located in a region where is uniform over the entire object.
c. the object is as large as the body that exerts the gravitational force on it.
d. any of the conditions above is satisfied.
e. only (a) or (b) above is satisfied.
(In my opinion it's b, but I can't give a good reason for it, please give explanation?)

Pairs of forces of equal magnitude act on identical cylinders as shown in the figures. In which example is the cylinder in translational and rotational equilibrium? (PLEASE EXPLAIN WHY)

Explanation / Answer

The center of gravity and center of mass are used synonymously in a uniform gravitational field to represent the unique point in an object or system which can be used to describe the system's response to external forces and torque.  Option b is true In the second part, the third object has both rotational and translational equilibrium -------------------------------------------------------------------------------------------------- The first ball inin translational equilibrium Second ball is in rotational equilibrium Fourth ball has both rotational and translational motion towards right Fifth ball has spinning motion so, only third ball is in rotational and translational equilibrium

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