Explanation

A person of mass m rides in a lift that accelerates at a, with upward positive. Newton’s second law gives the cable tension, or the scale reading, T = m(g + a).a = 0 is at rest so T is the true weight;a = −g is free fall.

The same equation covers a load hanging from the lift ceiling and a person standing on a scale. Both are the apparent weight.

Taking upward as positive:

• T − mg = ma
• so T = m(g + a)

If the lift accelerates upward, T is greater than the weight. If it accelerates downward, T is smaller. If a ≤ −g the cable slacks and T = 0.

Worked example

A person of mass 80 kg stands in a lift. Take g = 9.81 m/s². Find the scale reading when the lift is at rest.

1. T = 80 × 9.81 = 784.8 N, the true weight

The lift now accelerates upwards at 1.2 m/s². Find the scale reading.

1. T − 784.8 = 80 × 1.2
2. T = 784.8 + 96 = 880.8 N

The lift instead accelerates downwards at 1.2 m/s². Find the scale reading.

1. T = 80(9.81 − 1.2) = 688.8 N

The reading is greater than the weight when the lift accelerates up, and smaller when it accelerates down. In free fall, a = −gand T = 0.