Weight & Normal Force

Weight is the gravitational force Earth exerts on a body, W = mg, pointing toward Earth's centre. It is the one force in a mechanics problem you can write before knowing anything about the motion: a 70.0 kg climber has W = 687 N standing on a ledge, hanging on a rope, or in mid-fall. The g in it is a field strength, g = GM_E/R_E², whose natural unit is N kg⁻¹; since 1 N kg⁻¹ = 1 m s⁻², the same number is the free-fall acceleration. Near-Earth is an approximation with a known size. For a height h small compared with R_E, g drops by the fraction 2h/R_E: 0.31% at an airliner's 10 km, and 1% only at about 32 km. Earth's spin and flattening move the measured value from about 9.78 N kg⁻¹ at the equator to 9.83 N kg⁻¹ at the poles; this course uses 9.81 N kg⁻¹. And weight is not mass: on the Moon, where g = 1.62 N kg⁻¹, the same climber weighs 113 N with the same 70.0 kg.

Rest a book on a table and the table pushes up on it; press on the book and the table pushes harder. The push comes from compression. The surface's atoms are squeezed slightly closer together and, like an extremely stiff spring, push back perpendicular to the contact. The squeeze is usually invisible: a timber floor that sags 0.50 mm under a 70.0 kg person acts like a spring of stiffness 686.7 N ÷ (5.0 × 10⁻⁴ m) ≈ 1.4 × 10⁶ N m⁻¹, and concrete is far stiffer. That stiffness is what licenses the model. Treat the surface as rigid, stop tracking the compression, and replace it with a rule: the object does not move into the surface. The normal force is then whatever magnitude, zero or more, enforces the rule. Its direction is always perpendicular to the contact and away from the surface: up from a floor, sideways from a wall, down from a ceiling you press against. The part of the contact force along the surface is friction, a separate model.

Because N has no formula, it comes out of Newton's second law. Pick one axis along the surface normal n̂. Put every other force on the free-body diagram, take each one's component along n̂, and let the constraint supply the acceleration along n̂, which is zero for a flat surface that is not itself moving. N is then the only unknown in that equation. A 6.00 kg crate on a level floor pulled with P = 25.0 N at 35.0° above the horizontal gives N + P sin θ − mg = 0, so N = 58.86 N − 14.34 N = 44.5 N. Push down at 35.0° below the horizontal instead and N = 58.86 N + 14.34 N = 73.2 N. The horizontal part of P appears in neither equation: it cannot change N. Tilt the surface to 30.0° with no other force acting and the normal axis tilts with it, so only mg cos θ presses in and N = 0.866 mg.

No bathroom scale senses gravity. It measures how hard its platform must push up on whatever stands on it, which is the normal force, and displays N/g: a mass calibrated for someone standing still with nothing else acting. That displayed push is the apparent weight, and it parts company with mg whenever another vertical force acts or the support accelerates. A 60.0 kg student on a scale who pulls down on a fixed overhead bar with 150 N is pulled up by the bar with 150 N, so N = 588.6 N − 150 N = 438.6 N and the display reads 44.7 kg. Pushing up on a low ceiling with the same 150 N makes the ceiling push her down, and the display reads 75.3 kg. Her weight is 589 N throughout. With no other vertical force but a vertical acceleration a_y, the same equation gives N = m(g + a_y), the idea behind every elevator problem.

Surfaces push; without glue or suction they cannot pull. So N ≥ 0 is part of the model, and the algebra can break it. Increase the upward pull on the 6.00 kg crate and N = mg − P sin θ reaches zero when P sin θ = mg = 58.9 N: a 58.9 N pull straight up, or 118 N at 30.0°. Beyond that the formula returns a negative number. Never report it. A negative N says that the assumption fed into the equation, a_y = 0, is false. The floor has released the crate, so N = 0, and the vertical equation now fixes the acceleration instead: a_y = (P sin θ − mg)/m, which for 150 N at 30.0° is (75.0 N − 58.86 N) ÷ 6.00 kg = 2.69 m s⁻² upward. The same test decides the instant a jumper's feet leave the ground, and when a load is left behind by a support that drops faster than free fall.

The constraint model is safe because real surfaces compress by fractions of a millimetre. Where that stops being true, so does the model. On a trampoline the deformation is the physics: a bed that behaves like a spring of stiffness 5.0 × 10³ N m⁻¹, pressed 0.60 m deep, pushes with 3.0 × 10³ N, about six times the weight of a 50.0 kg gymnast, and that force changes from instant to instant, so N must be computed from the stretch rather than imposed through a⊥ = 0. With several supports, such as four table legs, two feet or four tyres, the second law gives only the total normal force. How it divides depends on where the supports sit, through rotational equilibrium later in the course, and with more supports than equations it depends on how legs and floor deform: a table with one short leg puts nothing on that leg, and rocks. And on a curved track or a hill crest the path itself demands an acceleration along the normal, so N ≠ mg even at constant speed with nothing else acting.