Weight & Apparent Weight
Weight is the gravitational force that the Earth (or any massive body) exerts on an object. It is a vector quantity, always directed towards the centre of the gravitating body. Unlike mass — which measures how much matter an object contains and never changes — weight depends on the local gravitational field strength. An astronaut has the same mass on the Moon as on Earth, but weighs roughly six times less because the Moon's gravitational field is weaker. This distinction between mass and weight is one of the most important ideas in Newtonian mechanics. Apparent weight is subtly different from true weight. It is the contact force — the normal force — that a surface exerts on you, which is what a bathroom scale actually measures. In an ordinary room, apparent weight equals true weight because you are in equilibrium. However, in an accelerating lift, on a roller-coaster, or in orbit, apparent weight can differ dramatically from true weight. Understanding why requires Newton's second law applied carefully to the forces acting on a body. These ideas have profound practical consequences. Engineers designing lifts, aircraft, and spacecraft must account for apparent weight changes. Pilots and astronauts train for high-g manoeuvres where apparent weight can multiply several times over, causing physiological stress. At the extreme, in free fall, apparent weight drops to zero — the condition called weightlessness — even though gravity is still acting on the object.
- True Weight
- Newton's Second Law and the Normal Force
- Free Fall and Weightlessness
- Apparent Weight Greater Than Real Weight
True Weight
True weight W is defined as the gravitational force on an object of mass m in a gravitational field of strength g: W = mg. On Earth's surface g ≈ 9.81 m s⁻². Because g varies slightly with altitude and latitude (it is slightly less at the equator than at the poles), true weight is not perfectly constant across Earth's surface, though for most problems we treat g as uniform.
Newton's Second Law and the Normal Force
Consider a person of mass m standing on a scale inside a lift. Two forces act: weight W = mg downwards, and the normal force N upwards (this is the apparent weight). Applying Newton's second law: N − mg = ma, where a is the acceleration of the lift. Therefore N = m(g + a). If the lift accelerates upward (a > 0), N > mg and you feel heavier. If it accelerates downward (a < 0), N < mg and you feel lighter.
Free Fall and Weightlessness
If the lift cable snaps and it falls freely, a = −g, so N = m(g − g) = 0. The scale reads zero — you are weightless. This is identical to what astronauts experience in orbit: the spacecraft and occupants are both in continuous free fall towards Earth (they just have enough horizontal velocity to keep missing it). Gravity is not absent — it is causing the orbit — but there is no contact force, so apparent weight is zero.
Apparent Weight Greater Than Real Weight
When a pilot pulls out of a dive, the aircraft accelerates upward. If the upward acceleration is 2g, then N = m(g + 2g) = 3mg. The pilot experiences three times their normal weight (3g). Fighter pilots wear special suits to prevent blood from pooling in the legs during high-g turns. Racing drivers on banked curves and passengers on loop-the-loop rides experience similar effects, where the normal force from the track or seat provides the centripetal acceleration plus supports weight.
The Key Equation
- N — Normal force (apparent weight) (Newton (N)): The contact force exerted by a surface on the object; this is what a scale measures. Positive upward.
- m — Mass (Kilogram (kg)): The amount of matter in the object; an invariant scalar that does not change with location or acceleration.
- g — Gravitational field strength (Metres per second squared (m s⁻²)): The acceleration due to gravity at the object's location; approximately 9.81 m s⁻² at Earth's surface.
- a — Acceleration of the frame (Metres per second squared (m s⁻²)): The vertical acceleration of the enclosing system (lift, vehicle, etc.). Positive upward; zero for stationary or constant-velocity motion.
Worked Example
Problem. A student of mass 60 kg stands on a bathroom scale inside a lift. The lift accelerates upward at 2.5 m s⁻². (a) What is the student's true weight? (b) What does the scale read (apparent weight)? Take g = 9.81 m s⁻².
Answer. True weight = 588.6 N; Apparent weight = 738.6 N (scale reads approximately 75.3 kg).
Origins · 1 minIsaac Newton and how this was discovered
Newton formulated the law of universal gravitation in his 1687 masterwork Principia Mathematica, showing that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. He also developed the three laws of motion, which together explain why weight causes acceleration and how apparent weight arises in accelerating frames. Newton reportedly said the idea that gravity might extend to the Moon came to him while observing an apple fall in his garden during the plague year of 1666, when Cambridge University was closed and he was working at home in Woolsthorpe.
Related Concepts
- Newton's Second Law — Apparent weight is derived directly by applying F = ma to the forces on a body in an accelerating frame, making Newton's second law the essential mathematical tool for this topic.
- Gravitational Field Strength — The value of g determines true weight at any location, and understanding how g varies with altitude and planetary mass is fundamental to calculating weight in different contexts.
- Newton's Law of Gravitation — Universal gravitation underpins the concept of weight as a gravitational force and explains why g differs on different planets and at different distances from a mass.
- Free Fall & Projectile Motion — Free fall is the limiting case where apparent weight becomes zero, connecting weightlessness to the kinematics of objects accelerating under gravity alone.