how does gravity work in a lift?

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Solution

Case 1: When the lift is at rest or moving with constant velocity (zero acceleration). In this case the normal force acting on the man in upward direction will be equal to the weight of the man ( $m g$ ) acting in downward direct due to the gravity,
$F_{N} = m g$

Case 2: When the lift is accelerating in upward direction. In this case normal force will act in upward direction, the weight of the man will act in downward direction and the acceleration of the lift/man is in the upward direction. Therefore, here using Newton's second law of motion we can say that force acting on man due to the accelerated lift is,
$F = m a$
Therefore the normal force or the apparent weight of the man will be given as,
$F_{N} = m g + m a$

Case 3: When the lift is accelerating in downward direction. In this case, normal force will act in upward direction, the weight of the man will act in downward direction and the acceleration of the lift/man is in the downward direction. Similarly, here the force acting on man due to the accelerated lift is,
$F = m a$
Therefore the normal force or the apparent weight of the man will be given as,
$F_{N} = m g - m a$

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