I'm having a lot of trouble with QUESTION 14 (shown in the picture). I feel as though my conceptual understanding is off because i dont agree with the answers. The answer sheet is attached.
My confusion lies with the line: "Radially: $T \cos \alpha = mv^2/r$"
I disagree with this! I feel as though they're messed up the cosine ratio. Isn't T the adjacent side, and the centripetal force acting as the hypotenuse? i feel as though it should be:
$\cos \alpha = \frac{T\cdot r}{mv^2}$ (after simplification)—which is very different to the answer shown.
You're implicitly considering the inertial reference frame of the common centre of circular motion. From that reference frame, the forces acting on the skier on the flat sea level plane of motion are only two: tension of the perfectly horizontal rope $T$ and drag due to air, $kv^2$. The weight of the skier and the reaction from the water, etc. are irrelevant as these forces are perpendicular to the flat plane of motion and are exactly balanced (otherwise the rope would not remain horizontal).
The key is that tension $T$ is the primary driving force of the skier's circular motion. It has to be resolved into tangential and radial components of force. The tangential component is exactly balanced by the aerodynamic drag (because of the constant tangential speed of the skier). The radial component is wholly responsible for keeping the skier in constant circular motion, i.e. it is equal to the centripetal force.
As the primary driving force, the tension $T$ automatically becomes the hypotenuse of the force triangle. The resolved components (radial and tangential) are necessarily strictly less than the tension. This is like doing a vector sum in reverse (you can also think of the tension as the resultant of the tangential and radial forces).
This is why the expressions are written as they are. The primary driving force is always the hypotenuse when you resolve into perpendicular force components.
One more thing: the centripetal force should always be viewed as the result of another (driving) force. It doesn't arise "de novo". A body cannot remain in constant circular motion if it is not being supplied a centripetal component of force by some driving force (which can be tension in a rope, weight (gravity), a contact reaction force or other forces, depending on the physical scenario). It does not make sense to try to resolve the centripetal force (i.e. don't treat it as the hypotenuse).