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

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.

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:
  Fg = G \frac{m1 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).
- Fg \propto m1 \text{ and } Fg \propto m2 (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.

Applies when surfaces are not sliding relative to each other.
Inequality: 0 \le fs \le \mus 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).

Applies when surfaces slide relative to each other.
Magnitude is constant for given (\muk) and (N): fk = \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.

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.

For particles/discrete masses:
x{cm} = \frac{\sum mi xi}{\sum mi},\quad y{cm} = \frac{\sum mi yi}{\sum mi},\quad z{cm} = \frac{\sum mi zi}{\sum mi}

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.

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).

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.

\vec F = m\vec a (introduced formally with Newton’s Second Law in later section).
Fg = G \frac{m1 m_2}{r^2}
F_g = m g
0 \le fs \le \mus N
fk = \muk N
\vec a_{avg} = \frac{\Delta \vec v}{\Delta t}
\vec a = \frac{d\vec v}{dt}

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).