Although infinity is a rather illusory and abstract concept, and it is quite taboo in physics to use any definition of "infinite", it is still a quite interesting question. Let's not talk about "infinitely long" ladders for now. Perhaps it is more acceptable to use "long enough".
If the ladder is long enough, there is no need to consider whether the material of the ladder can withstand such a large mass. We are just doing thought experiments and letting our imagination run wild.
Before conducting the thought experiment, you need to briefly understand the knowledge of cosmic speed .
We all know that there is gravity between everything, especially for massive celestial bodies, the gravity is even more obvious. For example, it is the gravity of the earth that binds us and allows us to stand safely on the earth. There are two ways to overcome the earth's gravity and leave the earth.
The first and most direct way is to have a long enough ladder as in the question, and then you climb up the ladder until you reach the Earth's gravitational range (theoretically the Earth's gravitational range is infinite, but in practice There are many other celestial bodies on the planet, and this kind of infinity does not exist).
The second method is to use the concept of cosmic speed. If the speed is fast enough, it can move in a circle along the earth without falling off. This is what we often call the first cosmic speed .
The first cosmic speed is about 7.9 kilometers per second.
But this speed is not enough for you to leave the earth. If you want to leave the earth, you need to increase the speed to reach the second cosmic speed, which is 11.2 kilometers per second. At this time, you can escape from the earth and escape the earth's gravitational range.
The problem in the question is to use the first method. There is a concept in physics: Hill sphere , which is simply the gravitational range of celestial bodies. Except for this range (Hill sphere), even if it leaves the gravitational constraints of celestial bodies, it is not controlled by celestial bodies.
The calculation formula for the radius of Hill sphere is: r≈a³√m/3M, a represents the radius of the celestial body, m is the mass of the object, and M is the mass of the celestial body. According to this formula, the radius of the Earth's Hill sphere can be calculated to be 1.5 million kilometers.
In other words, when you are 1.5 million kilometers away, you are no longer bound by the earth's gravity. However, if the height of the ladder reaches 36,000 kilometers (geostationary orbit height), the linear speed of the exceeded part of the ladder will be greater than the orbital speed, which means that the exceeded part of the ladder will cause the ladder to be completely damaged due to centrifugal tendency, and you will die as a result. . If the ladder is very, very strong and does not destroy, then only the earth will be destroyed.
So, the result is that no matter how long the ladder is, you can't escape from the earth.
This is why when humans launch satellites and various detectors, they will choose to use speed to escape the earth's gravity, because once the speed is reached, no power is needed to leave the earth.
This is like when we ride a bicycle uphill. We must speed up before going uphill, so that we can go uphill easily without much effort. If you don't speed up before going uphill, you can still go uphill, but it will take a lot of effort. First, you have to make sure you have enough strength.
