Forces, Gravitation, Friction, Mass–Weight, and Acceleration

Fundamental Ideas About Forces

• Force ((\vec F))
  • Vector quantity experienced as a push or pull.
  • SI unit: newton ((\text N)).
    $1\,\text N = 1\,\text{kg}\cdot\text{m}\,\text{s}^{-2}$
  • Can act with contact (e.g.
    pushing a box) or at a distance (e.g.
    gravity, electrostatic forces between charges).
• Connection to Acceleration
  • Any change in velocity (i.e.
    acceleration) results from one or more forces acting on an object.
  • Upcoming sections will combine these ideas into Newton’s Laws and translational equilibrium.

Newton’s Law of Universal Gravitation

• Historical motivation
  • Newton noticed apples always fall toward the Earth’s center (not sideways) → wondered if the same attractive force reaches the Moon.
• Law & Equation
  • Magnitude of gravitational force between two point masses:
    $F<em>g = G \frac{m</em>1 m_2}{r^2}$
    where
  • (G = 6.67 \times 10^{-11}\,\text{N m}^2\,\text{kg}^{-2}) (universal gravitational constant)
  • (m1, m2) = masses
  • (r) = distance between centers of mass.
• Proportionality insights
  • $F_g \propto \frac{1}{r^2}$ (halve (r) ⇒ force quadruples).
  • $F<em>g \propto m</em>1 \text{ and } F<em>g \propto m</em>2$ (triple a mass ⇒ force triples).
• Scope & Significance
  • All matter exerts gravity on all other matter.
  • On small scales (you vs. a book) gravitational forces are usually negligible compared with other forces; at planetary scales they dominate.
• Worked Example
  • Find (Fg) between a proton ((mp = 1.67 \times 10^{-27}\,\text{kg})) and an electron ((me = 9.11 \times 10^{-31}\,\text{kg})) separated by (r = 1.0 \times 10^{-11}\,\text{m}). Fg = G \frac{mp me}{r^2}
    = (6.67 \times 10^{-11})\frac{(1.67 \times 10^{-27})(9.11 \times 10^{-31})}{(1.0 \times 10^{-11})^2}
    $\approx 1.0 \times 10^{-45}\,\text N$
  • Emphasizes how tiny gravitational attraction is at atomic distances compared with (for example) electrostatic forces.

Friction

Overview

• Always opposes relative motion (or attempted motion) between surfaces.
• Generates thermal energy; never accelerates objects in direction of motion.

Static Friction ((f_s))

• Applies when surfaces are not sliding relative to each other.
• Inequality: $0 \le f<em>s \le \mu</em>s N$ where
  • (\mu_s) = coefficient of static friction (unit-less, depends on material pair)
  • (N) = normal force (perpendicular contact force).
• Important consequences
  • Range of values; adapts up to a maximum threshold (f{s,\max} = \mus N).
  • An object can remain at rest under applied forces below this threshold.
    • Example: suitcase resists 25 N & 50 N pushes, only moves when (\gtrsim 100\,\text N) → implies (50 < f_{s,\max} < 100\,\text N).

Kinetic Friction ((f_k))

• Applies when surfaces slide relative to each other.
• Magnitude is constant for given (\muk) and (N): $f</em>k = \mu_k N$
• Key distinctions from static friction:
  1. Equality sign → single value, independent of speed or contact area.
  2. Always smaller than (f{s,\max}) because (\muk < \mu_s) for any material pair.
• Clarifying example: A rolling wheel experiences static friction (no sliding at contact point). Only when it skids (e.g.
  on ice) does kinetic friction act.

Practical / MCAT Reminders

• Pay attention to wording: “neglect friction” or provided (\mus, \muk) values.
• Friction appears in force-balance (translational equilibrium) problems.

Mass vs. Weight

• Mass ((m))
  • Measures quantity of matter / inertia.
  • Scalar; SI unit: kilogram (kg).
  • Independent of location (Earth vs. Moon).
• Weight ((\vec F_g))
  • Gravitational force on mass by a planetary body.
  • Vector; calculated by
    $F_g = m g$
    where (g \approx 9.8\,\text{m s}^{-2}) (often rounded to 10 (\text{m s}^{-2})).
• Concept of Center of Mass / Gravity
  • Weight can be modeled as acting at a single point.
  • Location depends on mass distribution.
  • For uniform, symmetric objects → geometric center; for irregular bodies → determined by weighted coordinate averages.

Center of Mass (3-D Formula)

For particles/discrete masses:
$x<em>{cm} = \frac{\sum m</em>i x<em>i}{\sum m</em>i},\quad y<em>{cm} = \frac{\sum m</em>i y<em>i}{\sum m</em>i},\quad z<em>{cm} = \frac{\sum m</em>i z<em>i}{\sum m</em>i}$

• Demonstration: Thrown tennis racquet’s various parts trace complicated trajectories, but its center of mass follows a simple parabola like a single tossed ball.
• MCAT seldom asks directly for these coordinates but may require recognizing/using the concept in multi-step mechanics questions.

Acceleration

• Definition: Rate of change of velocity due to applied force.
• Vector; SI units: (\text{m s}^{-2}).
• Average acceleration:
  $\vec a_{avg} = \frac{\Delta \vec v}{\Delta t}$
• Instantaneous acceleration (limit as time interval → 0):
  $\vec a = \lim_{\Delta t \to 0} \frac{\Delta \vec v}{\Delta t} = \frac{d\vec v}{dt}$
• Graphical interpretation
  • On a velocity–time plot, the slope of the tangent at time (t) gives instantaneous acceleration.
  • Positive slope ⇒ acceleration in same direction as velocity.
  • Negative slope ⇒ acceleration opposite velocity (often called deceleration).

Conceptual & Ethical/Practical Connections

• Scale dependence: Recognizing when forces (e.g.
  gravity vs.
  friction) dominate is crucial for modeling real systems—from atomic to astronomical.
• Implications of friction
  • Engineering: brake systems rely on maximizing (\mus) and (\muk).
  • Energy: friction converts mechanical energy into heat → efficiency concerns.
• Universal gravitation
  • Foundation for orbital mechanics, satellite technology, and understanding tides.
  • Philosophically illustrates the universality of physical laws (same formula describes apples, moons, and galaxies).
• Safety considerations: Understanding static vs.
  kinetic friction helps predict when objects will slip—a key factor in workplace and transportation safety.

Quick Reference (Formulas)

• $\vec F = m\vec a$ (introduced formally with Newton’s Second Law in later section).
• $F<em>g = G \frac{m</em>1 m_2}{r^2}$
• $F_g = m g$
• $0 \le f<em>s \le \mu</em>s N$
• $f<em>k = \mu</em>k N$
• $\vec a_{avg} = \frac{\Delta \vec v}{\Delta t}$
• $\vec a = \frac{d\vec v}{dt}$

Study Tips for the MCAT

• Memorize the form and proportionalities of Newton’s gravitational law; expect “what happens to (F_g) if…” questions.
• Differentiate clearly between mass and weight; convert properly when moving between Earth and other planets.
• Always verify if an object is static or sliding before selecting (fs) vs. (fk).
• Draw free-body diagrams: include weight at center of gravity, normal force, friction, and any applied forces to visualize acceleration conditions.
• Practice reading v-t graphs quickly to interpret slopes (acceleration) and areas (displacement).