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 Homework Statement
 A solid hemisphere is placed on a horizontal ledge so that its center is directly above the edge, fixed there by a delicate thread. It is given a push of negligible velocity and begins to topple over the edge. When it collides perfectly inelastically with the vertical face of the ledge, the thread breaks. What is the kinetic energy at this moment?
 Homework Equations

CM of a hemisphere: [itex]\frac{3}{8}R[/itex]
Moment of inertia about geometric center: [itex]\frac{2}{5}MR^2[/itex]
Moment of inertia about CM: [itex]I_0=\frac{83}{320}MR^2[/itex] (can be derived via parallelaxis theorem)
Angular momentum: [itex]J=I_0\omega+Mv_\mathrm{CM}r_\mathrm{CM}[/itex] (where [itex]r_\mathrm{CM}[/itex] is relative to the point about which angular momentum is measured)
Kinetic energy: [itex]K=\frac{1}{2}Mv_\mathrm{CM}^2+\frac{1}{2}I_0\omega^2[/itex]
I emphasise again that this is a problem I created myself to better my understanding of moments of inertia and angular momentum.
Does my latest approach seem reasonable? If not, can Approach 1 be salvaged?
Thanks,
QM
My latest approach: The method of successive approximations  Separate the hemisphere into a part with positive horizontal velocity and a part with negative horizontal velocity. Arrest the negative horizontal velocity of the lower piece. Then, "reconnect" the pieces in a way that conserves linear and angular momentum, and then repeat until there is no negative horizontal velocity portion. Now, I am expecting that this process should take infinitely many iterations to reach the answer, since the reasoning of Approach 2 seems like it will always produce a portion of the sphere with negative horizontal velocity. That I'm prepared to deal with. I'm expecting the piece with negative horizontal velocity gets smaller each time due to the increase in horizontal momentum of the hemisphere at each step; however, the angular momentum would then also decrease, though by a smaller amount each time. In any event, my instinct suggests it should not go to zero. This leads me to conclude the angular velocity of the top piece should remain the same before "reconnecting." (If it were set to zero, at a much later step, the hemisphere would be nearly stationary after collision, which seems nonphysical.)Approach 1: I recognise immediately that energy, momentum, and angular momentum are all nonconserved during the collision. My first strategy was to use bruteforce integration to determine "how much negative momentum" was in the quartersphere that had negative horizontal component of momentum. I then demanded that all of this negative horizontal momentum be erased in the collision, leaving an equal (and opposite, of course) momentum in the forward direction. This gave me a velocity of [itex]\frac{3\pi}{64}R\omega_0[/itex] where [itex]\omega_0=\sqrt{\frac{15g}{8R}}[/itex] is the angular velocity immediately before the collision. Next, I specified that immediately after the collision, the lowest point of the hemisphere should have no horizontal velocity due to the inelasticity of the collision. Being careful to realise that there are both horizontal and vertical components to the linear velocity produced by rotation about the CM, I conclude that the final angular velocity is [itex]\omega_1=\frac{3\pi}{64}\omega_0[/itex]. The problem is that I require the velocities of the upper quartersphere to redistribute throughout the hemisphere and I am concerned this affects the result; moreover, I should be able to apply similar "arresting" reasoning to the angular momentum of the hemisphere about its geometric center and cannot figure out my limits of integration. If I demand that only a quartersphere of angular momentum is arrested by the ledge, then I need a good reason why the domain of applicability does not extend further; there's no [itex]\sin\theta[/itex] factor here and thus no reason that is apparent to me why I shouldn't keep going. However, integrating over the whole hemisphere means it completely stops rotating, and that just doesn't seem instinctively physically accurate. (My instinct here is that neglecting the angular momentum effects here means my linear momentum is lower than it should be.)
Approach 2: I dissect the hemisphere into two identical quarterspheres and demand that, immediately after the collision, neither quartersphere is rotating, the upper quartersphere maintains its original momentum vector, and the horizontal velocity of the lower quartersphere is arrested. I then imagine that the quarterspheres momentarily "reconnect," in a way which conserves momentum and angular momentum (but not kinetic energy; c.f. two blocks colliding perfectly inelastically). Doing this gives me [itex]v_1=\frac{3}{8}R\omega_0[/itex] and [itex]\omega_1=\frac{45}{166}\omega_0[/itex]. Here you can already see the problemthe linear velocity of the lowest point of the hemisphere has negative horizontal component!
Does my latest approach seem reasonable? If not, can Approach 1 be salvaged?
Thanks,
QM