Frame of reference in elevator

Question

I have a conceptual question that I have trouble understanding.

An object at the end of a rope is attached to the ceiling of an elevator. If the elevator is allowed to free-fall and hit the ground, at impact the tension of the rope is greater than if the elevator was standing still.

I would like to know why in the equation below, the acceleration is positive for $m\vec{a}$

$$T - mg = ma$$

Is this because to the object at the end of the rope, the elevator appears to be accelerating upward? Why is the acceleration not negative, since the elevator was free-falling?

Answer 1

The object at the end of the rope has a negative velocity (in the world frame of reference) when the elevator (and it) comes to a halt.

When the elevator stops, you have to go from a negative velocity to zero velocity. That requires a positive acceleration.

Answer 2

You're probably used to thinking about the objects at the bottom of ropes pulling downward on ropes because of gravity. But here, it's better to think of the top of the rope pulling up. This is because here's the sequence of events when the bottom of the elevator hits the ground: 1) The ground exerts a large upward force on the elevator floor in order to quickly stop its free-fall. 2) The elevator floor exerts a large upward force on the elevator's rigid walls. 3) The walls exert a large upward force on the elevator ceiling. 4) The ceiling exerts a large upward force on the top of the rope, creating tension. 5) This tension accelerates the object upward.

If the elevator parts are massive, they will absorb some of the upward force, but the rest will be transmitted. So basically, the ground is "pushing up" on the object at the end of the rope, but doing so in a slightly circuitous way that ends up creating a pulling effect.