天同師兄dc motor問題,同埋potentiometer

As space here is limited, and you posted so many questions. I could only give short answers.
Q1:You don't have to mind too much about the term sin(wt). The equation is used in most physics texts just illustrate the principle in practical motors. You would feel better if you could consider a radial magnetic field (instead of a uniform magnetic field) such that sin(wt) always equals to unity irrespective of the position of the rotating coil. Hence, back emf = NBAw and is always constant for a given rotational speed w.
Q2: When the coil is in a vertical position (aasume a uniform magnetic field acting horizontally), there is no back emf according to the formula. But don't forget that there is also no torques acting on the coil at this isntant, though a current may exist. The resultant current in the coil is in a sinusoidal form. But this has no practical significance, because the max current occurs at coil position (vertical) where there is no torque, and zero current occurs at coil position (horizontal) when the back emf is max and that also produces no torque.

Q3: There is forces acting on the coil at this position, but these forces produce no torque, as they are passing through the axis of the coil.

Q5: The heart of a potentiometer is the driving cell, which provides a steady current through the potentiomete wire, hence establishing a linear potential drop along the wire. The current from the cell under test is assumed small so as not to disturb the steady and linear potential drop along the wire as established by the driving cell. If this condition could not be met, the theory on potentiometer would not hold.

Q6&7: The cell (at bottom) is connected in a reverse direction. It is not a potentiometer circuit. Here, the internal resistance of the cell is importance, which absorbs the difference in potential drop , and cannot be negleced in the calculation.

If you still have queries, please pm me direct as space is not enough here.