Dynamics of Sliding Irregularly Shaped Objects - A Folded Pizza

Question

A pizza is folded twice until it resembles a sector of 1/4 the pizza area. It lies on the pan with a vertex at the original center position of the pizza, and a curved edge at the position of the original rim of the pizza. It must be given an impulse so that the center of mass of the folded pizza stops at exactly the original center of the pizza. Obviously this may be done by applying an impulse at the exact center of the folded edge. However, taking into account the possibility of applying a non-normally directed impulse (so that the slice spins into position as well as translates), 1) may the impulse be applied all the way to the edge of the curved sector, or is it restricted to a smaller region, and 2) does the zone of restriction depend on the coefficient of sliding friction of the pizza/pan surface? Assume the pizza loading is uniform across the slice, and the area density is also uniform.

If you need some food while you're solving this problem, you should be able to simplify the original conditions, and still be able to perform experiments.

Answer 1

At first, we need to make one additional assumption, namely value of the impulse is constant for any applied angle. Then we may consider the situation.

For any angles less than 90⁰, the applied force will be resolved into || and ⊥ components. F_⊥ will create rotation around CM, while F_|| will create translation motion. That means we have two restriction of the applied impulse P. The first related to the angle of P, and the second to value of it, as EK=EK_translation + EK_rotation.

The larger the angle, the less part of the applied force will be spent for the translational motion. Then, the larger the angle, the greater should be the applied force, in order to reach the original location.

 As for friction between the pizza and the pan, if it may be ignored, object will have only translational motion, and then see previous statements.

 If friction cannot be ignored the greater friction, the smaller the angle value, as the greater part of EK will be spent for rotation and for work against the friction.

Thus, in case of constant impulse applied, the angle should be close to the center of the folded edge