What does it mean that the weight is negative? In a recent question on the Aviation Stack Exchange, one explained that would cause an autogyro's crash. Basically, it says that the one thing that would make the gyro crash is to unload the motor too much, as explained here:

[...] Unfortunately, as soon as the rotor stops spinning, the whole aircraft falls like a brick and the rotor may be impossible to restart in flight. This is a situation that should be avoided at all costs.
Normally it is not a problem since the weight of the aircraft keeps the rotor spinning. However, if the weight becomes too low or even negative, the angle of attack will become negative, and the rotor will slow down and eventually stop. It can happen when the pilot "pushes on the stick" and dives.

That's this bold text that caught my attention. I learned in high school that the weight of an object equals its mass in kg times the acceleration of gravity in N/kg. An object mass cannot be negative and AFAIK, neither does the acceleration of gravity. So how can it be possible that the weight is negative?
 A: The weight of an object is the force given by Newton's second law:
$$ F = ma $$
As you say the mass is constant, and if an object is neither accelerating up nor down the acceleration is just the gravitational acceleration $g \approx 9.81 \text{ms}^{-2}$. Then we get the familiar equation for the weight:
$$ W = mg $$
However suppose you are doing a loop. If you've ever done this you'll remember that your weight increases at the bottom of the loop and decreases at the top of the loop. That's because at the bottom of the loop the plane is accelerating upwards and at the top of the loop the plane is accelerating downwards. If we call the plane's acceleration $a_p$ then the total acceleration is:
$$ a = g + a_p $$
and the weight becomes:
$$ W = m(g + a_p) $$
At the top of the loop the plane is accelerating downwards so $a_p \lt 0$ and that means $W \lt mg$ i.e. your weight is reduced.
Though I've never done it I'm told it is possible to pull a loop so tight that you experience negative acceleration at the top of the loop i.e. the plane's acceleration downwards is greater than the gravitational acceleration upwards and your weight becomes negative. I would guess this is what the post on the Aviation SE means.
