all about Newton's laws of motion

Newton’s Laws of Motion – The Complete Pack

1. The Three Laws in One Breath

1. 1st (Inertia) – A body stays at rest or in uniform motion unless a net force acts.

2. 2nd (Acceleration) – F = m a ; force = mass × acceleration (vector form F = dp/dt).

3. 3rd (Action–Reaction) – Forces come in pairs: equal magnitude, opposite direction, same line.

2. Words → Equations

- Inertia: if ΣF = 0 → v = constant.

- F = m a → acceleration ∝ force, ∝ 1/m.

- Action–Reaction: F₁₂ = –F₂₁ (never cancel because they act on different bodies).

3. Key Concepts Hidden Inside

- Inertial mass (m) measures resistance to acceleration.

- Net force (vector sum) matters, not individual forces.

- Acceleration and force are co-linear; instantaneous.

- Pairs act on different objects → don’t add them for one-body FBD.

4. Free-Body Diagram (FBD) Recipe

1. Isolate the body.

2. Draw all real forces (push, pull, weight, normal, friction, tension, lift, drag, magnetic, electric…) acting on that body.

3. Choose axes (tilt if needed).

4. ΣFₓ = m aₓ , ΣFᵧ = m aᵧ , ΣF_z = m a_z .

5. Solve algebra → numbers.

5. Common Force Formulae

- Weight: W = m g (g ≈ 9.81 m s⁻² downward).

- Normal: N ⊥ surface; magnitude adjusts to prevent penetration.

- Friction: fₛ ≤ μₛ N , fₖ = μₖ N opposes relative motion.

- Spring: F = –k x (Hooke; negative sign = restoring).

- Tension: along string/rope; same magnitude both ends if massless & frictionless pulley.

- Drag: F_d = ½ C ρ A v² (quadratic, opposes v).

6. Working the 2nd Law – Worked Templates

a. Horizontal pull – no friction

FBD: → F , ← 0

ΣFₓ = F = m a → a = F/m.

b. Block on inclined plane (angle θ, friction μₖ)

Choose x-axis // plane.

ΣFₓ = m g sin θ – μₖ m g cos θ = m a →

a = g (sin θ – μₖ cos θ).

c. Atwood machine (massless string, ideal pulley)

System equation: (m₁ – m₂)g = (m₁ + m₂)a →

a = g (m₁ – m₂)/(m₁ + m₂); T = m₁(g – a).

d. Circular motion (string, vertical circle)

Top: T + m g = m v²/r → T = m(v²/r – g).

Bottom: T – m g = m v²/r → T = m(v²/r + g).

7. Circular Motion & Newton

- Centripetal force is not a new force; it is the resultant of real forces pointing to the centre: ΣF_in = m v²/r.

- Direction continuously changes → acceleration even at constant speed.

8. Apparent Weight / Elevator

- Scale reads normal force N.

- Upward a: N = m(g + a) → heavier.

- Downward a: N = m(g – a) → lighter.

- Free-fail a = g → N = 0 (weightlessness).

9. Friction Revisited

- Static > kinetic: μₛ > μₖ.

- Impending slip: fₛ,max = μₛ N.

- Angle of repose: tan θ_max = μₛ.

10. Spring–Mass System

- F = –k x → a = –(k/m)x = –ω²x (SHM).

- ω = √(k/m), T = 2π√(m/k).

11. Momentum Language

- 2nd law