Answer:
(a) The elevator is traveling upward and its upward velocity is decreasing as it nears a stop at a higher floor. T > mg
(b) The elevator is traveling upward and its upward velocity is increasing as it begins its journey towards a higher floor. T > mg
(c) The elevator is traveling downward and its downward velocity is decreasing as it nears a stop at a lower floor. T < mg
(d) The elevator is traveling downward at a constant velocity. T = mg
(e) The elevator is traveling downward and its downward velocity is increasing. T < mg
(f) The elevator is stationary and remains at rest. T = mg
Explanation:
To answer this question, consider all the forces acting on the elevator.
The mass of the ball acting downwards due to gravity = mg
The tension on the string depends on upward or downwards force on the ball. T = m(a+g)
where a is acceleration and increase in velocity causes increase in acceleration, and vice versa. (a = v/t)
(a) The elevator is traveling upward and its upward velocity is decreasing as it nears a stop at a higher floor.
If the upward velocity is decreasing, its acceleration is also decreasing, and acceleration is not equal to Zero
T = m(a+g) > mg
(b) The elevator is traveling upward and its upward velocity is increasing as it begins its journey towards a higher floor.
If the upward velocity is increasing, its acceleration is also increasing.
Then, T = m(a+g) > mg
(c) The elevator is traveling downward and its downward velocity is decreasing as it nears a stop at a lower floor.
If the downward velocity is decreasing, its acceleration is also decreasing, and acceleration is not equal to Zero
T = m(a-g) < mg
(d) The elevator is traveling downward at a constant velocity
At constant velocity, acceleration is zero, because acceleration is the rate of change of velocity.
T = m(0+g) = mg
(e) The elevator is traveling downward and its downward velocity is increasing
If the downward velocity is increasing, its acceleration is also increasing
T = m(a-g) < mg
(f) The elevator is stationary and remains at rest.
if the elevator is at rest, its acceleration is zero
T = m(0+g) = mg
to 
(from (-a/2) to
(a/2))