To determine the speed of the turntable as a function of time, we need to consider the concept of conservation of angular momentum. However, without knowing more about the mass and rate of the syrup, we cannot determine the speed of the turntable as a function of time.
To determine the speed of the turntable as a function of time, we need to consider the concept of conservation of angular momentum. The initial angular momentum of the turntable is equal to the final angular momentum of the syrup being dropped onto it. The initial angular momentum of the turntable is given by the product of its moment of inertia and initial angular speed. The final angular momentum is given by the product of the moment of inertia of the syrup and its final angular speed.
Since the syrup is dropped in a circle onto the turntable at a certain radius, it will have its own moment of inertia. As the syrup is dropped continuously, the total moment of inertia of the system will change. In order to find the speed of the turntable as a function of time, we would need more information about how the mass of the syrup changes over time as well as the rate at which it is dropped.