Question

If tension in the cable supporting a lift moving downwards is half the tension when it is moving upwards the acceleration of the lifts

Answer 1

When acc. is downward, let tension be T. Therefore, tension in the string when the lift is accelerating upwards is 2T.

So when the lift is moving down. . . a = (T - mg)/m.
and when it is going up,. . . a= (2T-mg)/-m
= (mg - 2T)/m. .
Hope it works. . :-)

Answer 2

Given:

Tension in the cable supporting a lift moving downwards is half the tension when it is moving upwards.

To Find:

Acceleration of the lift

Solution:

Let tension in the cable when lift moves upward is T

Tension in the cable when lift moves downward = .5T

Free body diagram of lift when it moves upwards.

Tension T (upward)

weight = mg (downward)

net force = mass×acceleration

$T-mg = ma (1)$

Free body diagram of lift when it moves downwards.

Tension 0.5T (upwards)

weight = mg (downward)

$mg - 0.5T = ma (2)$

substitute T = mg + ma from equation 1 in equation 2;

$mg - 0.5(ma+mg) = ma\\.5mg = 1.5ma\\a = \frac{g}{3}$

Acceleration of Lift is equal to  one third of gravity.