Chapter 5 Lecture: Apparent Weight and Weightlessness

Foundations of Apparent Weight and Sensation of Motion

• Definition of Apparent Weight:

  • Apparent weight is defined mechanically as the magnitude of the supporting contact force—predominantly the normal force ($n$)—exerted upward on an object by a supporting surface, such as a floor or a weighing scale.

  • A standard weighing scale does not directly measure an object's true gravitational force ($w = mg$); instead, it measures the normal contact force that it exerts upward against the object to support it.

• Kinematic Sensations in an Elevator:

  • When an elevator initially transitions from a stationary rest state ($v = 0$) and accelerates upward toward a constant upward velocity, the passenger experiences a distinct physical sensation of being heavier.

  • Conversely, when the elevator starts from rest and accelerates downward, the passenger experiences a physical sensation of feeling lighter.

  • Once the elevator ceases accelerating and reaches a constant velocity (zero acceleration, where $\frac{dv}{dt} = 0$), the apparent weight immediately returns to the true gravitational weight, regardless of whether the elevator is traveling upward or downward.

  • Quality elevator engineering aims to minimize the magnitude and abruptness of this transitional acceleration to reduce physiological discomfort. In older or substandard elevator systems (commonly encountered in university buildings), these acceleration phases are abrupt and strongly felt.

Free Body Diagrams and Subscript Notation Conventions

• Coordinate System Protocol and Rigorous Sign Conventions:

  • An explicit coordinate system must be established before writing Newton's second law equations: the upward vertical direction is defined as positive ($+y$), and the downward vertical direction is defined as negative ($-y$).

  • Coordinate definition is an indispensable step for governing the mathematical signs of vector magnitudes. Omission of an explicit coordinate definition on handwritten problem sets results in a mandatory one-point deduction, as the coordinate frame is the sole objective guide determining whether forces and accelerations are entered with positive or negative signs.

• Point Particle Idealization:

  • For the purpose of free body diagrams, an individual standing on both feet is modeled as a single point particle of total mass $m$.

  • Internal distributions of contact between individual feet and the surface are consolidated into a single net normal force vector acting at the center of mass.

• Standardized Double-Subscript Notation:

  • Forces are denoted using a systematic two-subscript convention, $F_{\text{agent, object}}$, where the first subscript designates the entity exerting the force and the second subscript designates the entity experiencing the force.

  • $w_{\text{E, U}}$: The downward gravitational force exerted by the Earth ($\text{E}$) on you ($\text{U}$).

  • $n_{\text{S, U}}$: The upward normal contact force exerted by the scale ($\text{S}$) on you ($\text{U}$).

  • $n_{\text{U, S}}$: The downward contact force exerted by you ($\text{U}$) on the scale ($\text{S}$).

  • Note: In this notation, the letter "$\text{E}$" represents the Earth, not the elevator.

Mechanics of the Upward-Accelerating Elevator

• Newton's Second Law Formulation for Upward Acceleration:

  • Consider a passenger of mass $m$ accelerating upward with an acceleration of magnitude $a$ (directed in the positive $+y$ direction).

  • The two collinear forces acting along the vertical axis are the upward normal force $n_{\text{S, U}}$ and the downward gravitational weight $w = mg$.

  • Applying Newton's second law along the vertical axis:         $\sum F_y = n_{\text{S, U}} - w = m a$

• Algebraic Derivation of Apparent Weight:

  • Isolating the upward normal force:         $n_{\text{S, U}} = w + m a$

  • Substituting the fundamental gravitational weight equation $w = mg$:         $n_{\text{S, U}} = m g + m a$

  • Factoring out the mass term yields the final governing expression:         $n_{\text{S, U}} = m (g + a)$

• Physical Interpretation:

  • Because the acceleration $a$ is strictly positive, the total effective upward acceleration experienced by the passenger is $(g + a) > g$.

  • Consequently, the normal force $n_{\text{S, U}}$ is quantitatively larger than the gravitational weight $mg$. On a free body diagram, the upward vector arrow representing $n_{\text{S, U}}$ must be drawn visibly longer than the downward vector arrow representing $w$.

  • The physiological sensation of being heavier occurs because the scale is actively driving upward against the passenger's inertia, effectively "smashing" the passenger into the surface of the scale.

Scale Operation Mechanics and Newton's Third Law

• Internal Mechanics of Weighing Scales (Hooke's Law):

  • Mechanical scales operate internally on spring systems described by Hooke's law. When an external force compresses the internal spring, the spring exerts an equal and opposing restoring force determined by its structural spring constant.

  • The internal spring compresses until the force exerted by the passenger down on the spring is exactly balanced by the restoring force pushing back up.

  • Modern digital or electronic scales operate on the identical underlying principle using structural metal elements (strain gauges). Applied downward force bends the metal, and the resulting strain or pressure change is converted into a calibrated digital display readout.

• Application of Newton's Third Law (Action-Reaction Pairs):

  • The force exerted by the scale upward on the person ($n_{\text{S, U}}$) and the force exerted by the person downward on the scale ($n_{\text{U, S}}$) constitute an authentic Newton's third law action-reaction pair.

  • These forces act on different bodies and have precisely equal magnitudes:         $|n_{\text{S, U}}| = |n_{\text{U, S}}|$

  • A scale reads the exact magnitude of the force applied directly onto it ($n_{\text{U, S}}$). Because of the third-law equivalence, this registered value matches $n_{\text{S, U}}$.

Mechanics of the Downward-Accelerating Elevator

• Constancy of True Gravitational Weight:

  • An object's true gravitational weight ($w = mg$) depends strictly on its intrinsic mass $m$ and the local gravitational field strength $g$.

  • Because the passenger remains at the surface of the Earth at an effectively fixed radial distance, $w$ is strictly invariant regardless of the elevator's velocity or acceleration.

• Newton's Second Law Formulation for Downward Acceleration:

  • When the elevator accelerates downward, the acceleration vector points in the negative vertical direction ($-y$), having a value of $-a$.

  • Summing the vertical forces in accordance with the established coordinate frame:         $\sum F_y = n_{\text{S, U}} - w = m (-a) = -m a$

• Algebraic Derivation and Sensation of Lightness:

  • Rearranging to solve for the normal force:         $n_{\text{S, U}} = w - m a$

  • Substituting $w = mg$:         $n_{\text{S, U}} = m g - m a = m (g - a)$

  • Because $a > 0$, the factor $(g - a)$ is strictly less than $g$, reducing the contact force ($n_{\text{S, U}} < mg$).

  • On the corresponding free body diagram, the downward gravitational force vector remains unchanged in length, but the upward normal force vector must be drawn noticeably shorter than the downward vector to ensure a net downward force.

  • The scale registers a reduced contact force, generating the subjective sensation of feeling lighter.

Mathematical Summary of Apparent Weight States

• Upward Acceleration ($a_y > 0$):

  • $\sum F_y = n - mg = +ma$

  • Apparent weight: $n = m(g + a)$

  • Physical sensation: The passenger feels heavier ($n > mg$).

• Downward Acceleration ($a_y < 0$):

  • $\sum F_y = n - mg = -ma$

  • Apparent weight: $n = m(g - a)$

  • Physical sensation: The passenger feels lighter ($n < mg$).

• Zero Acceleration ($a_y = 0$):

  • Occurs when the elevator is completely at rest or translating at a constant vertical velocity ($v = \text{constant}$).

  • $\sum F_y = n - mg = 0$

  • Apparent weight: $n = mg$

  • Physical sensation: The passenger feels their ordinary, baseline gravitational weight.

Maximum Acceleration and Structural Load Analysis

• Structural Failure Mechanics:

  • Elevating objects subjects supporting structural floors, cables, and packaging to elevated normal forces during upward acceleration.

  • Real-world analogy: Grocery bags frequently rupture when an individual rapidly accelerates them upward while lifting. To prevent tearing, lifting acceleration must be minimized, or the bags should be lowered with downward acceleration where normal forces remain low.

• Crate Elevator Problem Parameters:

  • Mass of the cargo crate: $m = 400\,\text{kg}$

  • Maximum floor support threshold before structural breaking: $n_{\text{max}} = 6000\,\text{N}$

  • Standard gravitational acceleration: $g = 9.8\,\text{m/s}^2$

• Baseline Static Calculation:

  • The static gravitational weight of the crate is calculated as:         $w = m g = (400\,\text{kg})(9.8\,\text{m/s}^2) = 3920\,\text{N}$

  • Because $3920\,\text{N} < 6000\,\text{N}$, the elevator floor safely supports the crate at rest or moving at a constant vertical velocity upward or downward.

• Directional Failure Mode:

  • Downward acceleration reduces the contact force ($n < 3920\,\text{N}$), which decreases structural stress on the floor.

  • Structural breaking occurs exclusively under upward acceleration, where $n = m(g + a)$ can exceed the $6000\,\text{N}$ structural threshold.

• Mathematical Derivation of Safe Acceleration Limit:

  • To prevent the elevator floor from breaking, the upward normal force must satisfy the inequality:         $n \le n_{\text{max}}$

  • Expressing $n$ via Newton's second law:         $n - w = m a \implies n = w + m a$

  • Establishing the inequality threshold:         $w + m a \le n_{\text{max}}$         $m a \le n_{\text{max}} - w$         $a \le \frac{n_{\text{max}} - w}{m}$

• Numerical Computation:

  • Substituting the given system parameters:         $a_{\text{max}} = \frac{6000\,\text{N} - 3920\,\text{N}}{400\,\text{kg}}$         $a_{\text{max}} = \frac{2080\,\text{N}}{400\,\text{kg}} = 5.2\,\text{m/s}^2$

  • To operate the elevator safely without collapsing the floor, the upward acceleration must not exceed $5.2\,\text{m/s}^2$.

Free Fall, Cable Snapping, and the Physics of Weightlessness

• Elevator Cable Severance Scenario:

  • If the hoisting cables of an elevator snap, the elevator car, the scale, and the passenger enter complete free fall.

  • All components of the system accelerate downward exclusively under gravity at an acceleration of $a = -g$.

• Mathematical Proof of Zero Apparent Weight:

  • Substituting $a = g$ downward into the downward apparent weight equation:         $n = m(g - g) = m(0) = 0$

  • The normal contact force drops identically to zero ($n = 0\,\text{N}$).

  • Because the scale and the passenger fall with the identical downward acceleration ($g$), the passenger ceases to exert any contact pressure on the scale surface, and the scale reads zero.

• Sensation of Weightlessness vs. True Gravitational Weight:

  • The physiological sensation of possessing weight does not stem from gravity directly; it is produced entirely by contact forces exerted by supporting surfaces (such as a floor, scale, or chair) pushing back against the body.

  • The colloquial term "weightlessness" is physically misleading: an individual in free fall is not weightless in a gravitational sense. The Earth continues to exert the full gravitational pull of $w = mg$.

  • Weightlessness denotes purely the absence of contact or normal forces ($n = 0$), not the absence of gravity ($F_g \ne 0$).

• Inertial Mass Determination in Free Fall:

  • Because standard scales cannot function when apparent weight is zero, an object's mass cannot be weighed gravitationally during free fall.

  • Mass in zero-gravity environments must be measured via its inertial properties using Newton's second law:         $m = \frac{F_{\text{net}}}{a}$

  • A known, calibrated net force $F_{\text{net}}$ is applied to the mass, its resulting acceleration $a$ is measured, and the ratio yields the invariant inertial mass.

Orbital Free Fall and Gravitational Mechanics in Spaceflight

• Orbital Mechanics as Perpetual Free Fall:

  • Astronauts aboard the International Space Station (ISS) experience continuous weightlessness because both the spacecraft and everything inside it are in a perpetual state of free fall toward Earth.

  • Every object inside the capsule falls toward the center of the Earth at the exact same rate of gravitational acceleration, precluding any sustained contact forces from forming between objects and the hull.

• Gravitational Field Strength at Orbital Altitudes:

  • Gravity in low Earth orbit is not zero. The gravitational acceleration $g$ at the altitude of the ISS is only slightly less than its sea-level value of $9.8\,\text{m/s}^2$, determined strictly by Newton's law of universal gravitation as a function of radial distance from the Earth's center.

• Motion and Restraint Systems in Orbit:

  • To remain stationary relative to the spacecraft interior, astronauts must anchor themselves mechanically to structural support bars.

  • Without mechanical restraints, any applied external force imparts an acceleration that causes an object or astronaut to drift at constant velocity relative to the hull until another external net force acts upon it.