Answer
The hair should be hanging upward.
Work Step by Step
The only forces acting on the hair are the gravitational force and the tension force. However, in order to remain in centripetal motion, the net force needs to be downward, meaning that the downward force must be at least as large as gravity. This means that tension must be downward. However, if the hair is going downward, the tension would be upward, so it is impossible for the hair to hang down as depicted in the cartoon.
