After today's physics class, which involved orbital mechanics, I began thinking about ways in which humans could affect the Earth's spin rate or its path around the Sun.
Jumping all at once:
If all humans congregated at one place on Earth (7+ billion people in one city, while maintaining social distancing, of course), and then jumped simultaneously, there would be some repercussions. The energy of all that mass shifting in a short time could lead to an Earthquake, for example. But, that is not the sort of effect I am interested in.
Would Earth's path around the Sun be affected? The answer is, surprisingly, not in the slightest. The problem is that we would eventually land back where we started. The net mass of the system consisting of Earth and us will not have changed. As we are part of the total system that is in orbit, the forces exchanged during both the jump and landing would be internal to that system. It is not possible to change the system's velocity without a force exchange with something external to the system. For example, an asteroid collision could have some small effect on the Earth's orbit.
Running all at once:
OK. So, jumping failed. Maybe by running, we can impact the planet's spin rate. Imagine that we (the human population) were to gather somewhere on the equator, like Singapore. We collectively decide that we wish to change the length of a day on this planet. We decide to run along the direction of the Earth's spin with the expectation that it might slow the rotation down (there are not enough hours in a day, they say).
With our first step, we propel ourselves forward (the Earth pushes us in the direction we move via static friction), so we impart an equal static friction onto the surface of Earth in the opposite direction. However, every subsequent time that our foot strikes the ground, it slows us down before speeding us up again. In fact, if we maintain our jogging speed, each step results in a net linear impulse of zero (on us and the Earth), which means that each step has zero net effect on the angular momentum of either.
It seems we suffer from the same problem as we did while jumping. Our initial acceleration from rest gives a tiny net angular impulse to Earth, but it will undo itself when we decelerate, just as our jump was only temporary in the previous scenario.
The only way to accomplish either of the intended effects (disrupt orbital path or spin rate) is to do something more permanent, like sending payloads to space. These do indeed impart small net impulses onto the Earth. I could calculate their magnitudes, but I don't feel like it.
Blowing up the planet:
Frustrated with our wasted efforts, we decide to blow the planet up from the inside. It splits into two halves. Each hemisphere will orbit the Sun, but the precise orbit of each half will depend on the direction in which the planet splits apart. Regardless, the Earth gets the last laugh... The center of mass consisting of each of the hemispheres will remain in the original orbit, because again, the explosion is ultimately an exchange of forces that are internal to the system.