Vector Addition

Factual Questions
1

While looking for an explanation on orbits I ran across this on the Bad Astronomy webpage. Please explain vector addition to me. I understandwhy the tidal force would less on the far side of the Earth, but why does the force go out. Wouldn’t the moon be adding more gravitational force toward itself, not away? Please help.

http://www.badastronomy.com/bad/misc/tides.html

Mods if this quote is too long, feel free to shorten it or delete it.

"The strength of gravity depends on the distance from the source. The closer you are, the stronger the “pull” you feel. The
Moon’s gravity acts on the Earth; but the diameter of the Earth is large enough in relation to the distance of the Moon that the
side of the Earth nearer the Moon feels the Moon’s gravity significantly more strongly than the side of the Earth away from the
Moon. If you could stand at the center of the Earth you would feel the Moon’s gravity somewhere between the two.

This part is tricky, and is the hardest part of this explanation to understand. A drawing of these forces looks like this:
–> ----> ------->
far center near
side of Earth side

where the arrows represent the force (and direction) of the Moon’s gravity on these three points of the Earth. Now, we measure
the gravity of the Earth relative to the center of the Earth; everywhere on the Earth, the center is “down”. In a sense, we see the
center of the Earth as “at rest”. It is mathematically correct to then subtract the force of the Moon on the center of the Earth from
the force felt on the near and far sides. This is called vector addition. If we do that, our diagram will look like this:
<- X ->
far center near
side of Earth side

(Note that this drawing is not meant to be exact, but just to give a feel for what’s happening).

Now we see that in this sense, the Earth is stretched by the difference in the Moon’s gravity across the Earth. We call this effect
“tides”. Tides are a differential force, that is, they result by the difference in the force of gravity between two points. "

2

The moon’s gravity isn’t just acting on the water; it’s also acting on the Earth itself. So while the tide on the near side of the Earth is caused by the water being sucked away from the planet, the tide on the far side of the Earth is being caused by the planet being sucked away from the water.

By analogy, imagine riding on a bus. If you’re standing in the aisle when the bus accelerates forwards, you feel pushed towards the back of the bus, not because anything is pushing you back but because the force pushing you forwards is less than the force pushing the bus forwards. So (force acting on you)-(force acting on the bus) is negative, and the result is a force that appears to be acting in the opposite direction.

The same thing happens with the tides. In this case, the moon acts on the Earth and the water, and all of it is (from the moon’s perspective) being pulled toward the moon. But (force acting on the water on the far side)-(force acting on the Earth) is negative, and so from the perspective of someone on Earth the force acting on the water on the far side appears to be pushing the water away from the Earth.