Laws of Motion

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Newton’s Three Laws of Motion — The Bedrock of Classical Mechanics:

Newton’s First Law (Law of Inertia): A body at rest stays at rest, and a body in motion continues in a straight line at constant speed, unless an external unbalanced force acts upon it. This law introduces the concept of inertia — the natural tendency of objects to resist changes in their state of motion. Mass is the quantitative measure of inertia.

Newton’s Second Law: F = ma, derived more fundamentally from momentum as F = dp/dt = d(mv)/dt. When mass is constant: F = ma. When mass varies (like a rocket losing fuel): F_ext = mdv/dt + v_rel dm/dt. The SI unit of force is the Newton (N = kg·m/s²).

Newton’s Third Law: Every action has an equal and opposite reaction. Forces always occur in pairs acting on different bodies — never on the same body. The action and reaction forces are simultaneous and of the same type (gravitational ↔ gravitational, contact ↔ contact).

Free Body Diagrams — The Non-Negotiable Skill:

Drawing an accurate FBD is the single most important skill in mechanics:

1. Draw a dot or box representing the isolated body
2. Sketch all external forces as arrows from the centre of mass outward
3. Label each force clearly (weight = mg, normal = N, tension = T, friction = f, applied = F)
4. Include a coordinate system — for inclined planes, tilt axes along and perpendicular to the surface
5. Never show forces the body exerts on other bodies

⚡ CUET Tip: A body is in equilibrium (at rest or constant velocity) if and only if ΣF = 0 in all directions. This gives N = mg cosθ for inclined planes and ΣF_x = 0, ΣF_y = 0 for planar problems.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Classes of Forces:

Field forces act through space without physical contact: gravity (mg, always downward on Earth), electrostatic forces (Coulomb’s law between charges), magnetic forces (on moving charges or current-carrying wires).

Contact forces arise from physical interaction: normal reaction (perpendicular to surfaces, points away from the surface), tension (along strings, ropes, rods, chains — always pulls the body), friction (parallel to surfaces, opposes relative motion or impending motion), spring force (F = -kx, restoring toward natural length).

Friction — Detailed:

For a body at rest on a surface, static friction f_s adjusts to exactly oppose any applied force up to its maximum value: 0 ≤ f_s ≤ f_s(max) = μ_sN. Once the applied force exceeds μ_sN, the body begins to slide and kinetic friction applies: f_k = μ_kN (typically μ_k < μ_s, so once sliding starts, friction drops).

⚠️ The angle of friction φ is defined by tan φ = f_s(max)/N = μ_s. The friction angle is the angle between the resultant of normal reaction and friction, and the normal reaction alone.

Pulley Systems:

Fixed pulley (ideal): Changes the direction of force but provides no mechanical advantage. The tension on both sides of a massless, frictionless pulley is equal. Mechanical advantage = 1.

Movable pulley: The load is supported by both strands of the string. For a massless, frictionless movable pulley: tension T supports half the weight, so T = W/2 to lift weight W. Mechanical advantage = 2.

Atwood’s machine: Two masses m₁ and m₂ connected by a string over a frictionless pulley. Acceleration a = (m₁ - m₂)g/(m₁ + m₂) directed toward the heavier mass. Tension T = 2m₁m₂g/(m₁ + m₂).

⚡ CUET Tip: Always write the constraint equation for an inextensible string. If the string goes over a pulley and connects two bodies, when one body moves down by x, the other moves up by x (for a single pulley system). For a system with multiple pulleys, count the number of supporting strands.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Pseudo Forces in Non-Inertial Frames:

When observing motion from an accelerating reference frame, Newton’s laws appear to break down — this is why an accelerometer in an accelerating car shows “acceleration forward.” To restore F = ma in a non-inertial frame, we introduce a fictitious pseudo force: F_pseudo = -m × a_frame.

Lift accelerating upward: In the lift’s frame, add a pseudo force downward = ma_lift. Apparent weight = m(g + a_lift). If the lift cable snaps and it falls freely (a_lift = g downward), apparent weight = m(g - g) = 0 — true weightlessness.

Lift accelerating downward: Apparent weight = m(g - a_lift). If a_lift = g (free fall), apparent weight = 0.

Turntable: A person standing at radius r from the centre of a rotating turntable experiences a centrifugal pseudo force outward = mω²r in the rotating frame. In an inertial (laboratory) frame, the floor provides a centripetal force mω²r inward.

Connected Bodies with Massless Pulley:

For a system of two blocks connected by a string over a frictionless pulley, write separate equations: Block 1 (mass m₁): T - m₁g = m₁a (taking downward as positive for m₁) Block 2 (mass m₂): m₂g - T = m₂a (taking upward as positive for m₂) Adding: (m₂ - m₁)g = (m₁ + m₂)a → a = (m₂ - m₁)g/(m₁ + m₂)

⚡ CUET Pattern: CUET physics frequently tests Newton’s laws with friction on inclined planes. A block on an inclined plane at angle θ: component of weight along plane = mg sinθ, normal reaction = mg cosθ. Maximum static friction = μ_s mg cosθ. The block remains at rest if mg sinθ ≤ μ_s mg cosθ, i.e., tanθ ≤ μ_s. Once θ exceeds the angle of friction, the block accelerates down the incline with acceleration a = g(sinθ - μ_k cosθ). These problems frequently appear in Section II of CUET with friction coefficients given numerically.