Lecture 12 – Newton’s First Law & Applications

Newton’s First Law (Law of Inertia)

• Statement: “An object at rest remains at rest, and an object in motion remains in motion with a constant velocity, unless acted upon by an external (net) force.”
  • In symbols: $\sum F_{\text{ext}} = 0 \; \Longrightarrow \; a = 0.$
  • The law distinguishes velocity (a state of motion) from force (a cause of change of motion).
• Inertia
  • Definition: The natural tendency of an object to maintain its current state of motion (either rest or constant-velocity motion).
  • Greater mass ⇒ greater inertia (harder to change its velocity).
• Mass vs. Force
  • Mass is a scalar, SI unit $\text{kg}$.
  • Force is a vector, SI unit $\text{N}$ (newton).
    $1\,\text{N}=1\,\text{kg}\cdot\text{m}/\text{s}^2.$

Is a Force Needed Along the Direction of Motion?

• Common misconception: “A forward force is required to keep an object moving.”
• Reality from Newton I:
  • If $\sum F=0$, an object already in motion keeps moving at constant $v$ even without a forward-acting force.
  • Example: A car coasting on a “perfectly” friction-free highway would maintain its speed after the engine is turned off.
  • Real highways still provide:
    • Driving (engine) force $F_D$ transmitted by static friction between tires and road.
    • Resistive forces (air drag, rolling resistance) $F_R$.
  • Constant-speed cruising occurs when $F<em>D = F</em>R$ ⇒ $\sum F = 0$ ⇒ $a = 0$.

Static (Mechanical) Equilibrium

• Condition: $a = 0$ in every direction ⇒ both component sums vanish:
  $\sum F<em>x = 0, \qquad \sum F</em>y = 0.$
• This applies to
  • Objects at rest, and
  • Objects moving with uniform straight-line motion.

Choosing a Coordinate System

• Two practical rules cited in the lecture
  1. If the direction of acceleration is known, take that direction as the $x$-axis (simplifies $F_x = ma$).
  2. If acceleration is unknown or zero, pick axes that make the trigonometry or algebra easiest (e.g.
  • Align $x$ with a surface or an applied force,
  • Align $y$ perpendicular to that surface).

Worked Example 1: Car in Steady Motion

• Situation: Car travels on level road at constant speed.
• Forces acting
  • Driving (engine) force $F_D$ forward (at tire–road contact).
  • Air/rolling resistance $F_R$ backward.
  • Weight $mg$ downward.
  • Normal force $N$ upward (from road).
• Equilibrium equations (because $a = 0$):
  $\sum F<em>x = F</em>D - F<em>R = 0 \;\Longrightarrow\; F</em>D = F<em>R,$$\sum F</em>y = N - mg = 0 \;\Longrightarrow\; N = mg.$
• Key implication: A forward force may exist, but its only role is to cancel the backward resistive force; net force is still zero.

Worked Example 2: Crate on a Rough Horizontal Floor, Pulled by a Force $F_a$ at Angle $\theta$

(Numbers were not provided; we keep everything symbolic.)

• Forces on crate (air resistance neglected):
  1. Applied pull $F_a$ at angle $\theta$ above the horizontal.
  2. Kinetic or static friction $F_f$ opposite the horizontal component of pull.
  3. Weight $mg$ downward.
  4. Normal force $N$ upward from floor.
• Choosing axes:
  $x$ horizontal, $y$ vertical (lecturer’s “makes the math easy” rule).
• Static-equilibrium conditions ($a=0$):
  $\sum F<em>x = F</em>a\cos\theta - F<em>f = 0 \;\Longrightarrow\; F</em>f = F<em>a\cos\theta,$$\sum F</em>y = N + F<em>a\sin\theta - mg = 0 \;\Longrightarrow\; N = mg - F</em>a\sin\theta.$
• Observations
  • Pulling upward (positive $\sin\theta$) reduces the normal force and therefore may reduce friction (if $F_f = \mu N$).
  • If F_a\sin\theta > mg the crate would start lifting off (lecture hinted: physically impossible in this static case unless the floor loses contact).

General Notes on Force Components (from blackboard algebra)

• Any force $\mathbf F$ can be expressed as
  $F<em>x = F\cos\phi, \qquad F</em>y = F\sin\phi,$
  where $\phi$ is measured from the positive $x$-axis.
• When dealing with inclined planes, many instructors rotate axes parallel/perpendicular to the surface to eliminate the need for $\sin/\cos$ in normal-force expressions.

Conceptual & Practical Implications

• Inertia explains seat-belt necessity: When a car stops abruptly, passengers keep moving at previous speed (no net external force acts on them if they’re not restrained).
• Engineering: Machinery parts designed to run at steady speed must counterbalance all resistive torques; net torque $=0$ despite large internal forces.
• Astronomy: Planets do not slow down in orbit because, in the direction of instantaneous velocity, gravitational force is perpendicular; tangential net force ≈ 0.
• Ethics/Safety: Recognizing that motion does not “need a push” reduces fuel use—engines should supply just enough force to cancel losses, not to “keep the car moving.”

Key Takeaways

• Zero net force ⇒ zero acceleration (but not necessarily zero velocity).
• Inertial mass quantifies an object’s resistance to acceleration.
• Careful force diagrams + appropriate axes turn word problems into solvable component equations $\sum F<em>x=ma</em>x, \; \sum F<em>y=ma</em>y.$
• Coordinate-choice skill is as important as algebra; always align axes to known or simplest directions.
• Static equilibrium problems end with $\sum F = 0$; dynamic problems introduce $ma$ on the side containing the non-zero net force.