Question

The apparent weight of a person in a lift moving downwards is half his apparent weight in the same lift moving upwards with the same magnitude of acceleration. So the acceleration of the lift is:
A. \[g\]
B. \[\dfrac{g}{4}\]
C. $\dfrac{g}{2}$
D. $\dfrac{g}{3}$

Answer 1

Hint: This is the question of Newton’s laws of motion. The pseudo force acts on the person due to motion in a non-inertial frame (Lift). So, The solution is given by conversion of non-inertial frame to inertial frame with the help of pseudo force.

Complete step by step answer:
As we know that when lift moves upward or downward, it accelerates or de-accelerates. Which denotes that the lift behaves as a non-inertial frame. So if we want to observe the phenomena inside the lift, we should convert this non-inertial frame to an inertial frame. Because we can apply Newton’s laws of motion only for inertial frames of reference.
To convert a non-inertial frame to an inertial frame, we use pseudo force on the person.
Apparent weight is the virtual weight of man that makes the sense of heaviness or weightlessness during the motion of lift. Apparent weight is given by-
Here ${W_g}$ is the weight of a person due to gravity And ${F_p}$ is pseudo force acting on a person.

From given question, we can write that-
${({W_{app}})_{downward}} = \dfrac{1}{2}{({W_{app}})_{upward}}$
When lift moves in downward direction, pseudo force acts upwards and when lift moves upward then the pseudo force acts downwards. The pseudo force is given by-
${F_p} = {m_{person}} \times {a_{\text{lift}}}$

So the solution for given condition in question is-
$
  {W_g} - F = \dfrac{1}{2}({W_g} + {F_p}) \\
   \Rightarrow mg - m{a_{\text{lift}}} = \dfrac{1}{2}(mg + m{a_{\text{lift}}}) \\
   \Rightarrow \dfrac{1}{2}mg = \dfrac{3}{2}m{a_{\text{lift}}} \\
  \therefore {a_{\text{lift}}} = \dfrac{g}{3} \\
 $
So, the acceleration of lift is $\dfrac{g}{3}$ that will satisfy the given condition.
Hence, the correct answer is option (D).

Note: Sometimes we don’t remember that the lift constructs a non-inertial frame and applies Newton’s laws to the question, which will end up wrong in this condition. So, always make sure that the given frame is inertial.