If total Mechanical Energy is conserved, can you say that angular momentum is also conserved?
Yes, but with the caution that for Angular Momentum one must conserve not just the speed of the moment but the other factors as well, to wit mass and radius.
It's the reason that when a spinning skater brings thier arms close to their body they spin faster, and when they want to exit the spin they move thier arms out wide to lower thier angular momentum to where they can capture the remainder in movement and skate to ice losses instead of just moving with thier arms in close and falling over.
That depends what you're considering part of your system. If you include enough objects that all forces are internal to your system, then the angular momentum of the system as a whole will be conserved, even if the mechanical energy isn't. However, if there are external forces acting on an object (even conservative ones) the object's angular (and linear) momentum need not be conserved.
Consider, for example, a skateboarder going back and forth on a frictionless half-pipe. Here, mechanical energy must be conserved, since the only forces are gravity and the normal force. However, the skateboarder's momentum will certainly not be conserved (either magnitude or direction). Now, that being the case, it's not too hard to see that angular momentum can't be conserved either. Since the skateboarder reverses direction, the angular momentum must also change direction. And, since this happens smoothly, angular momentum will vary continuously.
Now, if we were to include Earth as part of our system, the changes in its momentum and angular momentum would compensate for those of the skateboarder, leaving these both conserved (well, at least if we neglect the Earth's orbital motion).
In the case of the half pipe, wouldn't the angular momentum be converted into potential energy and possibly linear momentum if the sides of the half pipe are vertical for some fixed (non-zero) distance?
Yes. A better way of looking at this is that angular/linear kinetic energy are converted to potential energy. Since the forces involved are conservative, total energy (kinetic plus potential) is conserved.
All three quantities subject to conservation laws (linear momentum, angular momentum, and energy) are conserved in a closed system. Open the system up to external forces and one or more of these quantities may not be conserved.
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