What is Apparent weight?

Example 25.14 (The elevator)

A $70\,\mathrm{kg}$ passenger stands on a scale in an elevator of vertical acceleration $a_z$ (upward positive): $N - mg = ma_z$, so $N = m(g + a_z)$. Pulling up at $a_z = 1.5\,\mathrm{m}/\mathrm{s}^{2}$: $792\,\mathrm{N}$, heavy; braking near the top: $582\,\mathrm{N}$, light; steady cruise: $mg = 687\,\mathrm{N}$ — the first law; free fall, $a_z = -g$: $N = 0$, weightlessness on a scale.

Example 25.15 (Two blocks and a pulley)

A block $m_1 = 4.0\,\mathrm{kg}$ on a frictionless table is tied by a light rope, over an ideal pulley, to a hanging block $m_2 = 1.0\,\mathrm{kg}$. The rope transmits the same tension $T$ at both ends; the blocks share one magnitude of acceleration $a$. One second law each, along each motion: $m_1 a = T$ and $m_2 a = m_2 g - T$, so $a = m_2\,g/(m_1 + m_2) = 1.96\,\mathrm{m}/\mathrm{s}^{2}$ and $T = m_1 a = 7.8\,\mathrm{N}$ — smaller than $m_2 g = 9.8\,\mathrm{N}$, as it must be for $m_2$ to accelerate downward.

Example 25.16 (The flat curve)

A car of mass $m$ rounds a flat curve of radius $R$ at constant speed $v$. In the Frenet frame (Chapter 24) the acceleration is purely normal, $a_N = v^2/R$, aimed at the center. Vertically, $N = mg$; horizontally, the only centripetal force on offer is the static friction of road on tyres: $f = mv^2/R \leq \mu_s N = \mu_s mg$, so $v \leq \sqrt{\mu_s\, g R}$, whatever the mass. For $R = 90\,\mathrm{m}$, dry road ($\mu_s = 0.70$): $v_{\max} \approx 25\,\mathrm{m}/\mathrm{s}$ ($89\,\mathrm{km}/\mathrm{h}$); on ice the ceiling collapses — Exercise 25.13 banks the road; Exercise 25.10 swings the same projection on a wire.