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Let's analyze each scenario:
(a) When the plate starts to spin faster:
The piece of food experiences a nonzero tangential acceleration and a centripetal acceleration.
Tangential acceleration occurs because the rotational speed of the plate increases, causing the linear speed of the food to increase.
Centripetal acceleration occurs because the food must accelerate inward toward the center of rotation to maintain its circular motion.
(b) When the plate rotates at constant angular velocity:
The piece of food experiences a nonzero centripetal acceleration but no tangential acceleration.
Centripetal acceleration exists because the food must continuously accelerate inward to maintain its circular motion at a constant speed.
Since the angular velocity (and hence the tangential velocity) remains constant, there is no tangential acceleration.
(c) When the plate slows to a halt:
The piece of food experiences a nonzero centripetal acceleration but no tangential acceleration.
Centripetal acceleration persists because the food must decelerate inward to maintain its circular motion even as the plate slows down.
Since the plate slows to a halt, the angular velocity (and hence the tangential velocity) decreases, resulting in no tangential acceleration.