Apparent Weight & Free-Fall
When you stand on bathroom scales in a moving lift, you might notice the reading changes as the lift accelerates up or down — yet your actual mass never changes. This is the distinction between true weight and apparent weight. Your true weight is the gravitational force the Earth exerts on you: W = mg, always directed downward. Your apparent weight, however, is the normal force N that a surface exerts on you — and this is precisely what scales measure. Because Newton's second law connects net force to acceleration, any vertical acceleration changes N without changing mg. The concept becomes dramatically clear in free fall. Imagine cutting the cable of a lift — everything inside falls together with acceleration g downward. The floor no longer needs to push up on you to keep you stationary relative to it, so the contact force drops to zero. You feel completely weightless, yet gravity is still pulling you just as strongly as ever. Astronauts in the International Space Station experience this continuously: they are in a state of perpetual free fall around the Earth, which is why they float. Their true weight is roughly 90% of what it is on the ground, but their apparent weight is zero. Understanding apparent weight requires applying Newton's second law carefully to a chosen system. By treating the person as the object and identifying all forces acting on them — gravity downward and the normal force upward — we can derive a single formula that covers every scenario: stationary lift, accelerating lift, and free fall.
- Identify the forces on a person in a lift
- Rearrange for apparent weight
- The free-fall limit
- Sign convention matters
- Orbital free fall and apparent weightlessness
Identify the forces on a person in a lift
Consider a person of mass m standing on scales inside a lift. Two forces act vertically: the gravitational force mg downward (true weight) and the normal force N upward from the scales (apparent weight). The net upward force is N − mg. By Newton's second law, N − mg = ma, where a is the lift's acceleration taken as positive upward.
Rearrange for apparent weight
Solving for N gives N = m(g + a). This single equation covers three cases. (1) Stationary or constant velocity (a = 0): N = mg — the scales read true weight. (2) Accelerating upward (a > 0): N > mg — you feel heavier, pressed into the floor. (3) Accelerating downward (a < 0): N < mg — you feel lighter. A lift decelerating while moving upward also has a downward acceleration, so you feel lighter during that phase too.
The free-fall limit
Free fall is the special case where the only force acting is gravity — no surface contact exists, or equivalently a = −g. Substituting into N = m(g + a): N = m(g + (−g)) = 0. Apparent weight is zero. This is weightlessness in the purest sense. Note that the person's inertial mass and gravitational mass are unchanged; gravity still acts. The sensation of weightlessness arises solely from the absence of any supporting contact force.
Sign convention matters
Always define a positive direction first — upward positive is conventional. If the lift accelerates downward at 3 m s⁻², write a = −3 m s⁻². If it accelerates upward at 3 m s⁻², write a = +3 m s⁻². Confusing the sign is the most common error in exam questions. A useful check: if the lift cable snaps (free fall), N must equal zero — if your answer gives anything else, recheck your signs.
Orbital free fall and apparent weightlessness
Satellites and space stations orbit because gravity provides the centripetal acceleration needed for circular motion. Every occupant and the station itself accelerate identically toward Earth, so there is no relative acceleration between person and floor — the normal force is zero throughout the orbit. This is why astronauts float. It is sometimes called microgravity, but the gravitational field at 400 km altitude is still about 8.7 N kg⁻¹, roughly 89% of surface gravity.
The Key Equation
- N — Normal force (apparent weight) (Newton (N)): The contact force the surface exerts on the person; this is what a scale measures.
- m — Mass (kilogram (kg)): The amount of matter in the person; invariant regardless of acceleration or location.
- g — Gravitational field strength (m s⁻²): Acceleration due to gravity at Earth's surface, approximately 9.8 m s⁻².
- a — Vertical acceleration of the lift (m s⁻²): Positive upward, negative downward. Free fall corresponds to a = −g, giving N = 0.
Worked Example
Problem. A student of mass 60 kg stands on scales inside a lift. The lift accelerates downward at 2.5 m s⁻². Taking g = 9.8 m s⁻², find (a) the student's true weight and (b) their apparent weight as recorded by the scales.
Answer. True weight = 588 N; Apparent weight = 438 N
Origins · 1 minIsaac Newton and how this was discovered
Newton formulated the three laws of motion and the law of universal gravitation in his 1687 masterwork Principia Mathematica. His second law, F = ma, is the direct foundation of the apparent weight equation N = m(g + a). By distinguishing between the net force on a body and any individual force within that system, Newton provided the conceptual tools needed to analyse why a scale reading changes in an accelerating lift or vanishes entirely in free fall. Newton reportedly delayed publishing his gravitational theory for nearly 20 years partly because early estimates of the Earth's radius gave results that did not quite match — he waited until better measurements confirmed his predictions before going public.
Related Concepts
- Mass and Weight — Apparent weight can only be understood after distinguishing mass (invariant) from true weight (gravitational force mg), which is the focus of this foundational concept.
- Gravitational Field Strength — The value of g determines true weight and appears directly in the apparent weight formula; understanding how g varies with altitude is essential for orbital weightlessness.
- Newton's Second Law — The apparent weight equation N = m(g + a) is derived by applying F = ma to the vertical forces on a person, making Newton's second law the central mathematical tool.
- Orbital Motion and Satellites — Satellites and space stations are in continuous free fall around Earth, making orbital motion the most dramatic real-world example of sustained apparent weightlessness.