Mass, weight, apparent weight and weightlessness
Mass is the amount of matter; weight = mg is gravity's pull on it. An accelerating lift changes your apparent weight, and in free fall you feel weightless.
Mass vs weight
| Point | Mass | Weight |
|---|---|---|
| Meaning | Amount of matter in a body | Force with which the Earth pulls the body |
| Formula | m | W = mg |
| Kind of quantity | Fundamental quantity; scalar (size only) | Derived quantity; vector (size and direction, towards the Earth's centre) |
| SI unit | kilogram (kg) | newton (N) |
| Does it change? | Same everywhere | Changes with g: less on the Moon, more at the poles |
| Measured with | Physical (beam) balance | Spring balance |
Apparent weight and weightlessness
Apparent weight is the weight you feel. It is the force with which you press on the floor. It equals the reaction force (R) of the floor on you. When a lift is at rest, R = mg, so apparent weight equals true weight. When the lift speeds up or slows down, R changes. A body is weightless when its apparent weight is zero. This happens in free fall, when the body falls with acceleration g. Astronauts in orbit float because they and their spacecraft are falling freely around the Earth together. Gravity still acts on them.
Apparent weight in a lift
| Motion of the lift | Reaction R | What you feel |
|---|---|---|
| At rest, or moving with uniform velocity (a = 0) | R = mg | Apparent weight = true weight |
| Moving up with acceleration a | R − mg = ma, so R = m(g + a) | Heavier than usual |
| Moving down with acceleration a | mg − R = ma, so R = m(g − a) | Lighter than usual |
| Falling freely, cable broken (a = g) | R = m(g − g) = 0 | Weightless |
How to draw the lift diagram
- Draw a tall rectangle for the lift, with a small weighing machine on its floor.
- Draw a person standing on the machine.
- Draw an arrow mg pointing down from the person and an arrow R pointing up from the machine.
- Draw an arrow a beside the lift for its acceleration. Label the four sketches a = 0, upward a, downward a and a = g.
- Write the result under each sketch: R = mg, R = m(g + a), R = m(g − a), R = 0.
Weighing machine in a lift
A person of mass 50 kg stands on a weighing machine in a lift. Find the reading in newtons when the lift (i) is at rest (ii) moves up with acceleration 2 m s⁻² (iii) moves down with acceleration 2 m s⁻². Take g = 9.8 m s⁻².
- Given: m = 50 kg, g = 9.8 m s⁻², a = 2 m s⁻².
- (i) At rest: R = mg = 50 × 9.8 = 490 N.
- (ii) Going up: R = m(g + a) = 50 × (9.8 + 2) = 50 × 11.8 = 590 N.
- (iii) Going down: R = m(g − a) = 50 × (9.8 − 2) = 50 × 7.8 = 390 N.
490 N at rest, 590 N going up and 390 N going down. If the cable broke, the reading would be 0 N (weightless).
In the lift of a tall building you feel heavy as it starts going up and light as it starts coming down. Astronauts like Rakesh Sharma, the first Indian in space, floated inside their spacecraft because it was in free fall around the Earth.
2 marks: “Distinguish between mass and weight (any two points).” Mass: amount of matter, scalar, kg, the same everywhere. Weight: gravitational force mg, vector, newton, changes with place.
2 marks: “Why do astronauts feel weightless in an orbiting spacecraft?” They and the spacecraft fall freely around the Earth with the same acceleration, so the floor does not push on them; apparent weight is zero.
7 marks: “What is apparent weight? Explain the apparent weight of a person in a lift that is at rest, moving up, moving down and falling freely.” Apparent weight = reaction R; at rest R = mg; up with a: R = m(g + a); down with a: R = m(g − a); free fall: R = 0, weightless; labelled lift diagrams.
Up, plus; down, minus; falling, zero: R = m(g + a), m(g − a), 0.