Laws of Motion

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Laws of Motion — Key Facts for BUET Newton’s First Law: object at rest stays at rest, object in uniform motion stays in uniform motion unless acted upon by external force (inertia) Newton’s Second Law: F = ma; more fundamental: F = dp/dt = d(mv)/dt Newton’s Third Law: every action has equal and opposite reaction; action and reaction act on different bodies Free Body Diagram (FBD): isolate one body, show ALL forces acting ON it with arrows ⚡ Exam tip: FBD is the #1 skill in BUET mechanics — draw it for every Newton’s laws problem!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Laws of Motion — BUET Study Guide

Newton’s Second Law detail:

• F = ma for constant mass
• F = dp/dt = d(mv)/dt for variable mass
• For constant mass: a = F/m

Normal reaction: On flat surface: N = mg On inclined plane: N = mg cosθ (perpendicular to surface) Parallel to surface: mg sinθ causes motion down slope

Tension: Force exerted by string, rope, or rod

• Inextensible string: same tension throughout
• Massless string: T is same at both ends
• For connected bodies: tension is internal force

Friction:

• Static friction: f_s ≤ μ_s N; maximum f_s_max = μ_s N
• Kinetic friction: f_k = μ_k N
• μ_s > μ_k always
• Friction force opposes relative motion (or impending motion)

Pulley systems:

• Fixed pulley: changes direction only, MA = 1
• Movable pulley: MA = 2 (force is halved if frictionless)
• Atwood machine: two masses connected by string over fixed pulley

Constraint relations: For inextensible string: acceleration magnitudes are equal; if one goes up distance x, the other goes down distance x

Pulley with mass: For massless pulley: tensions on both sides are equal For massive pulley: tensions differ; use moment of inertia

Free Body Diagram method:

1. Identify all bodies in the system
2. Draw separate FBD for each body
3. Apply F = ma for each body
4. Write constraint equations for connected bodies
5. Solve the system of equations

** Newton’s Second Law on inclined plane:** For block on incline with friction:

• Along plane: mg sinθ − f = ma

• Perpendicular: N = mg cosθ

• If impending motion: f = μ_s N

• Key formula: F = ma; N = mg cosθ (on incline); f_k = μ_k N; T = same throughout massless string

• Common trap: In FBD, only show forces ON the body, not forces the body exerts ON others

• Exam weight: 2–3 questions per exam (8–12 marks); very high weight

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Laws of Motion — Comprehensive BUET Notes

Variable mass systems: For rocket expelling mass: F = ma_ext + v_rel dm/dt Thrust = −v_rel dm/dt (opposite to direction of ejection)

Pseudo forces: In non-inertial frame, appear to act on all masses: F_pseudo = −ma_frame Used in problems involving accelerating frames (lift, car turning)

Centripetal force: F_c = mv²/r = mrω² Direction: towards centre of circular path

Banking of roads: tan θ = v²/(rg) θ = angle of banking; v = speed; r = radius; g = acceleration due to gravity

Conical pendulum: T cos θ = mg; T sin θ = mv²/r; tan θ = v²/(rg) Period of revolution: T = 2π√(h/g) where h is effective length

Friction on curved surface: For no friction: N = mg cos θ With friction: N = mg cos θ ± f sin θ depending on direction

String constraint: For two masses m₁ and m₂ connected by inextensible string over pulley: a = (m₁g − m₂g)/(m₁ + m₂) if m₁ > m₂

Acceleration of wedge: For small angle wedge pushed horizontally, block slides down: a_relative = g sin θ / (1 + m/M) or similar depending on configuration

Normal reaction as variable: In vertical circular motion, at top of circle: N + mg = mv²/r At bottom: N − mg = mv²/r

Circular motion dynamics: For mass moving in circle of radius r at speed v: Centripetal acceleration = v²/r = rω² Required centripetal force = m v²/r = mrω²

Minimum speed at top of vertical circle: v_min = √(gR) where R is radius Below this speed, the object cannot complete the circle

Tension in rotating string: At top of vertical circle: T + mg = mv²/R At bottom: T − mg = mv²/R

Atwood machine with massive pulley: If pulley has mass I and radius r, and string doesn’t slip: a = (m₁ − m₂)g / (m₁ + m₂ + I/r²)

Impulse-momentum theorem: Impulse = change in momentum = ∫ F dt = Ft (for constant F) J = Δp = p₂ − p₁

Collision types:

• Elastic: momentum conserved, KE conserved
• Inelastic: momentum conserved, KE not conserved
• Perfectly inelastic: bodies stick together after collision

Coefficient of restitution: e = (relative velocity of separation)/(relative velocity of approach) e = 0 for perfectly inelastic; e = 1 for elastic

Recoil of gun: Initial momentum = 0; final momentum = 0 M_g v_g + M_p v_p = 0; v_p = −(M_g/M_p) v_g

Rocket propulsion: Thrust = v_e dm/dt (v_e = exhaust velocity relative to rocket) Net external force: F_ext = m dv/dt − v_e dm/dt

Friction angle: tan φ = μ_s where φ is angle of friction For equilibrium on inclined plane: tan θ ≤ μ_s for no sliding

Ladder friction problems: For ladder against wall with friction on both surfaces: At friction equilibrium: μ_s N₂ = N₁ and similar equations

• Remember: FBD is essential for every problem; tension same in massless string throughout; for inextensible string, accelerations of connected bodies are related; friction opposes motion or impending motion
• Previous years: “Find acceleration of two blocks m1=5kg, m2=3kg connected by string over pulley” [2023 BUET]; “Block slides down 30° incline with μ=0.3, find acceleration” [2024 BUET]; “Find tension in string during elevator accelerating upward at 2m/s² with 10kg mass” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Physics

• Mechanics (Laws of Motion, Work-Energy, Rotational) — very high weight
• Electrostatics and Current Electricity — very high weight
• Modern Physics (Photoelectric effect, Atoms) — high weight
• Heat and Thermodynamics — medium-high weight
• Optics (Reflection, Refraction) — medium weight

📝 Previous Year Question Patterns

• Laws of Motion: 2–3 questions per exam, 8–12 marks
• Common patterns: FBD problems, pulley systems, friction on inclined planes, circular motion
• Weight: very high — foundational topic

💡 Pro Tips

• Always draw FBD first — it’s the single most important skill
• For connected bodies, write acceleration constraint: if one moves distance x, the other moves related distance
• For pulleys, remember tension is same throughout massless string
• Friction force = μN always; perpendicular components must be resolved correctly
• For circular motion, centripetal force is the net radial force (not an extra force!)

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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