If earth were to rotate faster than its present speed, the weight of an object

If earth were to rotate on its own axis such that the weight of a person at the equator becomes half the weight at the poles, then its time period of rotation is (g = acceleration due to gravity near the poles and R is the radius of earth) (Ignore equatorial bulge)

How many times faster than its present speed the earth should rotate so that the apparent weight of an object at equator becomes zero ? Given radius of the earth = 6.37 xx 10^(6) m . What would be the duration of the day in that case ?

How many hours would make a day if the earth were rotating at such a high speed that the weight of a body on the equator were zero

At which angular speed the earth should rotates so that the apparent weight of an object at equator will become zero ?

If diameter of earth becomes half of its present value and its mass becomes four times its present value, how would the weight of any object on the surface of earth be affected ?

If the speed of rotation of earth about its axis increases, then the weight of the body at the equator will

How does the weight of an object vary with respect to mass and radius of the earth. In a hypothetical case, if the diameter of the earth becomes half of its present value and its ,mass becomes four times of its present value, then how would the weight of any object on the surface of the earth be affected?