Gravity, elasticity, weight and friction

Scales in a braking lift read more, even though the person has eaten nothing and their mass is unchanged. To understand the reading, we will examine what acts on what: Earth, a spring, the floor and the person.

Why does an apple fall towards Earth while Earth seems not to respond? Gravitation is mutual: all bodies with mass attract one another with equal forces. F = Gm₁m₂/r² shows how the force magnitude depends on both masses and their separation. Earth's enormous mass makes its acceleration almost imperceptible. This form applies to point particles or non-overlapping, spherically symmetric bodies: an arbitrary nearby rock cannot simply be treated as a sphere without checking.

Imagine moving two spheres twice as far apart. The force becomes four times smaller: what matters is the square of r, the distance between their centres, rather than the gap between their surfaces. Doubling either mass instead doubles the force. In SI, masses are in kilograms and r in metres; G ≈ 6,67·10⁻¹¹ Н·м²/кг². This universal constant sets the strength of gravitation. It is not the local acceleration g and does not change if the experiment moves to another planet.

How does gravitation relate to the measured acceleration of falling? Near a planet's surface, the gravitational force is Fтяж = mg. Doubling the mass doubles the force, but also doubles inertia: a = Fтяж/m = g stays the same. On Earth, g is about 9.8 m/s². On another planet, g and the gravitational force change while mass stays the same. We can treat g as constant over a region small compared with the planet's radius.

Hang a load from a spring: the spring extends and pulls it back. Deformation changes the relative positions of parts of the spring, and the elastic force tends to restore its shape. The extension Δl is the new length minus the original length, in metres. It is negative during compression, so the force magnitude is related to |Δl|. Find the direction separately: a stretched spring pulls towards its original length, while a compressed spring pushes.

Add identical small loads to the same spring one at a time. Within the linear model, doubling the load doubles the deformation: Fупр = k|Δl|. The spring constant k, in N/m, gives the force per metre of length change. At the same extension, a stiffer spring exerts a larger force. On a graph of Fупр against |Δl|, k is the slope of the straight line; at zero deformation, this model gives zero force.

Can we keep stretching a spring along the same straight graph? No: Hooke's law describes a limited linear range. Elasticity means recovering the original shape when the load is removed, but elastic behaviour need not be linear at every deformation. Excessive stretching breaks the proportionality, and a damaged spring may not recover. Before using k, therefore, check both the material and the load range.

A book sliding on a table slows down: the surfaces exert forces on one another along their contact. Kinetic friction opposes their relative sliding. The approximate model Fтр = μN relates its magnitude to the normal force N, perpendicular to the surface; μ is dimensionless. Within this model, doubling N usually doubles friction. N does not always equal mg: an extra pressing force or acceleration of the support changes the normal force.

Push a cupboard gently, then harder. While it remains at rest, the horizontal net force is zero: static friction adjusts to the push. For a 10 N push, it may be 10 N in the opposite direction; with no horizontal load, it may be zero. Only just before sliding does it reach a limit described using the coefficient of static friction and N. Saying that friction always equals μN therefore misses the important distinction between rest and sliding.

A person presses on the scales, and the scales press on the person. Here weight P means the force a body exerts on a support or suspension, measured in newtons. The normal force N acts back on the person: these contact forces form a Newton's-third-law pair. The gravitational force mg is a different force, exerted by Earth on the person. Even if P and mg have equal magnitudes, they act on different objects. The scale reading reflects contact, rather than directly measuring mass or gravitation.

To accelerate a person upwards, the support must both balance gravity and provide an extra upward force. Taking upward as positive, N − mg = ma, so the weight magnitude is P = N = m(g + a), where a is the magnitude of the upward acceleration. When a = 0, we get mg; increasing a increases the weight. The lift may actually be moving downwards and braking: its reading depends on the change in velocity, rather than the current direction of motion.

If the lift accelerates downwards with magnitude a < g, gravity partly provides this acceleration and the support presses less strongly: P = m(g − a). At the limit a = g, the person and floor fall freely together. No supporting normal force is needed: P = 0. Gravitation still acts. We cannot extend the formula to a negative force pressing on the floor: an ordinary support cannot pull a person down by the feet, and losing contact changes the model.

A person of mass 50 kg is travelling downwards. The lift brakes with an upward acceleration of 2 m/s². Take g = 10 м/с². Find the person's weight.

The lift moves downwards but brakes, so its acceleration of 2 m/s² points upwards. To produce this change in velocity, the support force must exceed mg.

N acts upwards on the person and mg acts downwards. Newton's second law gives N − mg = ma; the person's weight on the support has magnitude N.

P = m(g + a) = 50 · (10 + 2) = 600 Н.

At constant velocity the reading would be 500 N. The extra 100 N gives 100/50 = 2 м/с² upwards; the person's mass remains 50 kg.

The weight is 600 N, which is 100 N above its usual value for the chosen g. The increased pressing force is due to the lift braking, rather than a change in mass.

The same mass can experience different gravitational forces and have different weights: distinguish the interactions and the objects they act on.

Spring forces and friction follow approximate laws; the deformation range and whether sliding occurs matter.

Weight changes with the support's acceleration. In free fall the contact force vanishes, although gravitation still acts.

Gravitation and Newton's second law will help explain a satellite's orbit.