I also don't get why the boy must even be accelerating as wouldn't the yoyo also do this when going fast in general?
For the yoyo to stay in the tilted position when there is no acceleration, all forces must balance out. Newton's 1st law applies when $a=0$.
But the only forces present are gravity down and the string tension at an angle and they are definitely not balancing out in the horizontal direction. The string tension has a horizontal component that will accelerate the yoyo back towards the starting point.
(Only if air resistance or so is included in the situation will there be another horizontal force to balance out that string tension. This is what you see in real life - but air resistance is negligible at low speeds and thus usually not included in physical models.)
If the skater is accelerating just as much forward as the yoyo is accelerated horizontally by the string tension, then the yoyo will never catch up. It will look like it stays still in this tilted position while both it and the skater accelerate forward.
- If he suddenly accelerates a bit more, then he accelerates away from the yoyo - that will increase the tilting angle. This will in turn make the string more horizontal, increasing the horizontal component of string tension, thus increasing the acceleration until it is equal again.
- And vice versa if he slows down the acceleration, the tilting angle will decrease causing a decreasing acceleration until it again fits.
So clearly the angle is a direct reflection of the acceleration.
Now, how do you solve this...
This question from the textbook stumps me as it seems basic equations like $F=ma$ cannot be used (or at least lots of information is obscured) and it seems like there isn't enough information to do it.
You can easily use Newton's 2nd law, even though the amount of information feels limited. Never trust that feeling; never give up too early on Newton. Sometimes you don't need much.
- Draw the force diagram, set up Newton's 2nd law horizontally (and the 1st law vertically if needed) and see if you have any extra unknowns. If you have any, then find more equations (possibly in other directions) to cover for the extra unknowns. Repeat this until all extra unknowns are covered for. Then you can solve for the acceleration.
