The title of this post is a question often posed by flat earthers to challenge the idea of a round earth. Because of its association with flat earthers, this question is often laughed at. I find this to be a shame, because its a quite interesting question. It also gives me the chance to do math and show off how smart I am!
I am going to start by answering a similar but related question. The earth is orbiting the sun at 66,660 mph [1]. Why doesn't the sheer speed of that orbit leave us all behind in the vacuum of space? The answer is inertia. Inertia is described in Newton's First Law of Motion: "An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted upon by an outside force." How does that answer the question? Well, think about an airplane. Commercial airliners fly at about 600 mph. Do the passengers spend the entire flight pressed back into their seats at 600 mph? If one of them stepped out into the aisle, would he be propelled to the back of the airplane at 600 mph? No. Why? Because while the plane is moving at 600 mph, so are the passengers, the air in the cabin, and everything else in the plane. No additional force is required to keep them moving at that speed because of inertia. They will continue moving at 600 mph until acted upon by an outside force, i.e., the plane slowing down. In fact, if the plane were to vanish, they would continue moving at 600 mph until coming to an unpleasant stop due to the outside force of the ground. Military bombers have to account for this when they drop bombs. The bombs will keep moving at the speed of the airplane, so they have to calculate how far ahead of the target they need to be for their bombs to hit the target.
However, returning to the airliner, the passengers will be pressed back into their seats when the plane initially accelerates to 600 mph. This is because of Newton's Second Law of Motion, which states that Force = Mass * Acceleration. This tells us that force is only present when there is acceleration, and vice versa. So while the plane is accelerating, force is exerted on the passengers. While the plane is at a constant speed, no force is exerted on the passengers, and they are free to move about the cabin and eat disgusting airline food. Importantly, these effects are the same no matter what the speed is. A train traveling at 40 mph or a car traveling at 70 mph will have the same effects. This continues to be true no matter how large the number is. Inertia holds true whether you are traveling at one millimeter per millenia or 100,000 miles per second.
Returning to the question (finally!), we don't fly off the earth even though it is orbiting at 66,660 mph because it is traveling at a constant speed. If it were accelerating in some direction, we would feel a force. However, it is traveling at a constant speed, so we don't notice it.
Now, five paragraphs in, to the main topic. (Internet knights just love the sound of their own voice, don't they?) If the earth is spinning, that introduces a new consideration: rotation. Due to inertia, an object traveling in a straight line will continue traveling in a straight line, not a circle. So if the earth is spinning at 1000 mph, it should have left us behind long ago, right? Wrong. The big number of 1000 mph (1037.5646 if you want to be pedantic) looks impressive and scary. But when you do the math, it's anything but. Yes, the earth is spinning, but only once a day. That's 360 degrees per day, or 15 degrees per hour, or 0.25 degrees per minute, or 0.00417 degrees per second. So if we take your current velocity to be moving at an angle of 0 degrees, you are currently moving at 1037 mph at 0 degrees. Next second, you will be moving at 1037 mph at 0.00417 degrees, then at 0.00834 degrees, then at 0.01251 degrees, and so on. Notice what's happening? You're traveling at a constant speed, and your direction is barely changing. Going back to the plane example, if the pilot turned the plane 0.25 degrees every minute, would you notice? Not in the least.*
*Technically, you could if you put a level on the floor. Since planes bank in order to turn, you would notice the floor was slightly off level. However, my point is that you wouldn't feel your direction changing.
Intuitively, it's already clear why we don't fly off the earth. It's spinning so slowly that we can't even feel our direction changing, although we can measure it. But, for completeness sake, let's calculate how much acceleration we're all undergoing due to the earth's rotation. Let's also do the calculation in metric, because it's easier. The equation for centripetal force if acceleration = velocity squared / radius [2]. The earth has a radius of 6,378,000 meters, and it is rotating at 1,669.8 km/h or 463.83 meters / second. This gives us a centripetal acceleration of 0.0337 meters per second squared. That giant 1000 mph figure turned into a fraction of a fraction.
For the fun of it, let's find out how much force it would take to hold your average overweight American to the earth if gravity didn't exist. Suppose the overweight American weighs 100 kg. We can use Newton's Second Law of F=ma to find the answer of 3.37 Newtons, or 0.758 pounds. A quarter inch polyester ropecan hold 400 pounds, which is over 400 times stronger than needed [3]. So our conclusion is this: if gravity didn't exist, you could counter the acceleration of earth's rotation indefinitely using nothing more than string.
