A person is standing on a weighing machine placed on the floor of an elevator. The elevator starts going up with some acceleration, moves with uniform velocity for a while and finally decelerates to stop. The maximum and the minimum weights recorded are 72 kg and 60 kg, respectively. Assuming that the magnitudes of acceleration and deceleration are the same, find (a) the true weight of the person and (b) the magnitude of the acceleration. Take g = 9.9 m/s2.
Solution
Maximum weight will be recorded when the elevator accelerates upwards.
Let N be the normal reaction on the person by the weighing machine.
So, from the free-body diagram of the person,
\[N = mg + ma\] ...(1)
This is maximum weight, N = 72 × 9.9 N
When decelerating upwards, minimum weight will be recorded.
\[N' = mg + m\left( - a \right)\] ...(2)
This is minimum weight, N' = 60 × 9.9 N
From equations (1) and (2), we have:
2 mg = 1306.8
\[\Rightarrow m = \frac{1306 . 8}{2 \times 9 . 9} = 66 kg\]
So, the true mass of the man is 66 kg.
And true weight = 66 \[\times\] 9.9 = 653.4 N
(b) Using equation (1) to find the acceleration, we get:
mg + ma = 72 × 9.9
\[\Rightarrow a = \frac{72 \times 9 . 9 - 66 \times 9 . 9}{66} = \frac{9 . 9 \times 6}{66} = \frac{9 . 9}{11}\]
\[ \Rightarrow a = 0 . 9 m/ s^2\]
