1.4 Hour Day
If the length of days on Earth were only 1.4 hours long, humans would be flung off the side of the Earth.
Level 1: You may have experienced what is called centrifugal force before in a car rounding a bend. When this happens you get pulled to the side of the car away from the turn. In the same way that the car turns, the Earth spins. One day is how long it takes for the Earth to spin in a whole circle. If the length of the day is shorter, then the Earth has to spin faster to finish the whole circle in one day. The faster the Earth spins, the more we feel getting pulled away from the Earth, just like if we turn faster in a car, we feel pulled away more. When the day is 1.4 hours long, that is just enough for people to be pulled off the side of the Earth into space.
Level 2: When a car turns around a corner, it experiences a centripetal acceleration towards the "middle" of the turn. Newton's First Law of Motion defines inertia, which can be described as the resistance to acceleration. The force that we feel (but is not an actual force, we are just experiencing inertia) is referred to as centrifugal force which is a fictitious force. Centripetal acceleration, and therefore the centrifugal force is greater as we speed up around a curve. Increasing the tangential velocity, or the angular velocity of the Earth creates a greater centrifugal force.
An object loses contact with another when the normal force between the two objects is 0. Centripetal force is not an actual force, but a net force, meaning that the sum of other forces will always equate to the Centripetal force. On the Equator of the Earth, in a simplistic view, someone standing here will have gravity pulling down on the person towards the Earth and Normal Force pushing the person up away from the Earth. If gravity is such that all of it is equal to exactly the Centripetal Force directed towards the center of the Earth, the Normal Force must equal 0.
This leaves that if the Earth has an angular velocity great enough such that the centripetal force is equal to gravity, a person would be able to be flung off the Earth. This value happens when the period of the Earth's rotation, or the length of a day, is 1.4 hours.
Level 3: The same concept discussed in level two applies, such that the object will fall off the Earth if its normal force against the Earth is 0. This happens when the centripetal force and gravity are equal as shown on the right because a normal force will unbalance this equivalence. We then use the formula for the period to derive the number of seconds it takes to complete one rotation around the earth's axis and change it to hours.
Extra Insight: You may have realized that there was a large simplification about these calculations. They only work accurately at the equator. This is because when we define centripetal acceleration, the radius we use should be the perpendicular line from the axis of rotation to the point along the circumference of the circle. As you can see on the left, the radius of a higher latitude is shorter than that of the one at the equator. Trigonometry gives us that the new radius of centripetal motion is given by the radius of the Earth times the cosine of the angle of the latitude. We can now do the same calculations from step 3 using the new radius to obtain a general solution based on the angle of latitude. Note that the radius used in the gravity side of the equation is still the same because gravity acts towards a point at the center of the Earth.
