Mastering the Cup-and-Ball Physics Problem in First Semester Kinematics Class

[scozzy]

So, I'm in a first semester physics class (kinematics is the gist of what's covered), and my professors gave me a problem that, if I can solve it, will cover every major point that will be in the first exam (projectile motion, rotational motion, energy). I've been working on it for the past few days, but I keep getting stuck early on.

Homework Statement

If you're familiar with the cup-and-ball game, you have a cup... and a ball. The ball is connected to the cup by a string, and the objective is to swing the ball around to land it in the cup. I am trying to solve for the initial velocity, v_o, such that it lands in the cup (origin) [look at the diagram at the bottom for clarification. v_o is pointed down]. There are no hard numbers given. Of course, values can be thrown in, like R=0.5m, or m=0.02kg, but I've been trying to do it with just variables.

Homework Equations

Projectile motion
Rotational motion
Energy
The only 'equation' our professor ever gives us is ƩF=ma. We're expected to know how to derive the rest, so there's no memorization here. I can't really give you any equations like you might expect from other kinematics classes.

Side note on the equations and math bit, somebody who did this problem last semester told me synthetic division was involved somewhere. Exciting!

The Attempt at a Solution

Here's what I've written down so far. http://i.imgur.com/jJZ5fa0.jpg

You can see my force diagram in the middle, which I'm fairly sure is accurate. The vector resultant in the x-axis is the centripetal acceleration, which would mean T₁=(mv_o²)/2. Easy enough. And in the y axis, gravitational force is the only force acting upon the ball, thus (since my positive y is going down) acceleration = g. I was thinking, maybe I need more points? So I drew more force diagrams, derived the components, but nothing helped. Points I tried were at the bottom of the circular motion, and somewhere between the bottom and middle of the circular motion.

On the bottom was my attempt at finding the critical angle, which I think is the bulk of the problem here. As soon as I find the critical angle, it becomes a simple projectile problem.

I thought, since we aren't given a time, that's indicative of needing energy, right? So I laid out the equations (with GPE=0 at origin level), but now I'm stuck with three unknowns (v_o, the initial velocity) (v₂, the velocity at the critical angle) and (θ_c, the angle at which tension will become zero, AKA when the string goes slack, AKA there is no more centripetal force acting on the ball).

Lastly, here's some quick projectile equations derived from the critical point. http://i.imgur.com/KMFiYDA.jpg?1

I think something just isn't clicking? Like, I'm comfortable with projectile, rotational motion, and energy by themselves. But adding them all together is daunting. Any nudges in the right direction would be super helpful!

 

[rcgldr]

Normally getting the ball into the cup is done with an upwards linear motion to get the ball to move upwards, then to maneuver the cup to catch the ball near the peak of it's nearly vertical path.

 

[scozzy]

Normally getting the ball into the cup is done with an upwards linear motion to get the ball to move upwards, then to maneuver the cup to catch the ball near the peak of it's nearly vertical path.

We're assuming the cup's location remains static throughout the problem. Kind of like if the ball was held out horizontally, then hit down with some force to give it an initial velocity downwards.