Laws of Motion

🎯 Key Points

• F=ma (Second Law); action-reaction pairs act on DIFFERENT bodies, never cancel each other on the same body (Third Law)
• Static friction (f_s ≤ μ_s N) adjusts to prevent sliding; kinetic friction (f_k = μ_k N) is constant once sliding, and μ_k < μ_s
• Centripetal force F=mv²/r always points toward the centre - it's not a new force, just the net result of existing forces (tension, gravity, normal, friction)
• Banking angle: tan θ = v²/rg - designed so friction isn't needed at the design speed
• Impulse J = FΔt = Δp; momentum is conserved when net external force is zero

Newton's Laws

• First Law: An object remains at rest or in uniform motion unless acted on by an external force (inertia)
• Second Law: F = ma (net force = mass × acceleration)
• Third Law: Every action has an equal and opposite reaction

Friction

• Static friction: f_s ≤ μ_s·N (maximum just before sliding)
• Kinetic friction: f_k = μ_k·N (constant during sliding, μ_k < μ_s)
• Angle of friction: tan(λ) = μ

Circular Motion

• Centripetal acceleration: a = v²/r = ω²r
• Centripetal force: F = mv²/r (directed toward centre)
• Banking angle: tan(θ) = v²/rg
• Critical speed at top of loop: v = √(rg)

Impulse and Momentum

• Impulse J = F·Δt = Δp (change in momentum)
• Conservation of momentum: if ΣF_ext = 0, then Σp = constant

Free Body Diagrams (FBD) - The Method

• Isolate ONE body and draw only the forces acting ON it: weight mg (always downward), normal reaction N (perpendicular to the contact surface), tension T (along the string, away from the body), applied force, and friction f (along the surface, opposing relative motion/tendency).
• Choose convenient axes (for inclines, take axes ALONG and PERPENDICULAR to the slope), then write ΣF = ma separately along each axis.
• Never include forces the body exerts on others - an action-reaction pair acts on two DIFFERENT bodies, so the two never appear in the same FBD.
• Solve the resulting simultaneous equations for the unknowns (acceleration, tension, normal force, etc.).

Connected Bodies: Pulley, Lift, and Chain

• Bodies on a table joined over a pulley: for a hanging mass m₁ pulling a mass m₂ on a smooth horizontal surface, a = m₁g/(m₁ + m₂) and tension T = m₁m₂g/(m₁ + m₂).
• Atwood machine (two masses over a pulley): a = (m₁ − m₂)g/(m₁ + m₂); T = 2m₁m₂g/(m₁ + m₂).
• Apparent weight in a lift: N = m(g + a) when accelerating up, N = m(g − a) when accelerating down; N = mg at rest or constant velocity; N = 0 (weightlessness) in free fall (a = g).
• Inextensible string: all connected bodies share the same magnitude of acceleration, and tension is uniform in a massless string over a frictionless pulley.

Friction on an Inclined Plane

• On an incline of angle θ, the weight resolves into mg sinθ (down the slope) and mg cosθ (into the surface), so N = mg cosθ.
• Angle of repose (α): the maximum incline angle at which a block just stays at rest → tan α = μ_s. If θ ≤ angle of repose, the block does not slide.
• Block sliding DOWN a rough incline: acceleration a = g(sinθ − μ_k cosθ).
• Block pushed UP a rough incline (friction acts down-slope): retardation a = g(sinθ + μ_k cosθ).

Banking of Roads (With Friction)

• For a frictionless banked road, the safe speed is fixed: v = √(rg tanθ), i.e. tanθ = v²/(rg).
• With friction (coefficient μ), a RANGE of safe speeds is allowed: v_max = √[rg(tanθ + μ)/(1 − μ tanθ)] and v_min = √[rg(tanθ − μ)/(1 + μ tanθ)].
• On a flat (unbanked) road, the turn relies on friction alone: v_max = √(μ rg).

Pseudo Forces in Non-Inertial Frames

• Newton's laws hold directly only in inertial frames (non-accelerating). In an accelerating (non-inertial) frame, they can still be applied by adding a pseudo force = −m·a_frame on every body, opposite to the frame's acceleration.
• Example: inside a car accelerating forward with acceleration a, a passenger feels a backward pseudo force ma; a hanging pendulum settles at tanφ = a/g from the vertical.
• Pseudo force explains the "apparent weight" in a lift and the outward "centrifugal force" felt in the rotating frame of a turning vehicle - it is a bookkeeping force, not a real interaction.

🚀 JEE Advanced Edge

Pseudo force in non-inertial frames: When analysing motion from inside an accelerating frame (like a lift or accelerating car), add a pseudo force = −ma_frame (opposite to the frame's acceleration) to make Newton's laws apply in that frame. This is the fastest way to solve "apparent weight in a lift" or "pendulum hanging in an accelerating car" problems without switching to the ground frame.

Connected bodies / Atwood machine with pulleys: For two masses connected over a pulley, write F=ma for EACH mass separately (using the SAME tension T and SAME magnitude of acceleration a, since the string is inextensible), then solve the two equations simultaneously: a = (m₁−m₂)g/(m₁+m₂) for a simple Atwood machine.

Worked problem: A block of mass 2 kg on a rough incline (θ=30°, μ=0.5) - will it slide? Approach: Compare tan θ to μ: tan(30°)≈0.577 > μ=0.5? No, actually tan(30°)=0.577 > 0.5, so since tan θ > μ, gravity's component down the slope EXCEEDS the maximum available static friction, and the block DOES slide. (If tan θ ≤ μ, it would stay put.)

Conservation of Linear Momentum

• When the net external force on a system is zero, its total linear momentum remains constant: this follows directly from Newton's second law.
• For an isolated two-body system, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, so any momentum lost by one body is gained by the other.
• Recoil of a gun: the forward momentum of the bullet equals the backward momentum of the gun, so the gun recoils.
• Explosion of a stationary bomb: the fragments fly off with momenta that add vectorially to zero.
• Momentum conservation holds during collisions and explosions even when kinetic energy is not conserved.

Equilibrium of Concurrent Forces

• A particle is in equilibrium when the vector sum of all forces acting on it is zero, so its acceleration is zero.
• For coplanar forces, equilibrium requires the sum of components along both the x and y axes to be zero separately.
• Lami's theorem: for three concurrent forces in equilibrium, each force is proportional to the sine of the angle between the other two.
• The three forces in equilibrium can be represented in magnitude and direction by the three sides of a triangle taken in order.
• Equilibrium may be static (body at rest) or dynamic (body moving with constant velocity).

Inertia and Its Types

• Inertia is the inherent property of a body to resist any change in its state of rest or of uniform motion; it is measured by mass.
• Inertia of rest: a body at rest tends to stay at rest, e.g. dust flies off when a carpet is beaten.
• Inertia of motion: a moving body tends to keep moving, e.g. a passenger lurches forward when a bus stops suddenly.
• Inertia of direction: a body resists change in its direction of motion, e.g. mud flies off tangentially from a spinning wheel.
• Greater mass means greater inertia; this is the basis of Newton's first law, also called the law of inertia.

Common Forces in Mechanics

• Weight is the gravitational force mg acting vertically downward on a body.
• The normal reaction N is the contact force exerted by a surface perpendicular to it.
• Tension is the pulling force transmitted along a string or rope; for an ideal massless string it is the same throughout.
• The spring force obeys Hooke's law, F = -kx, directed so as to restore the spring to its natural length.
• Contact forces (normal and friction) arise from electromagnetic interactions between surfaces, while weight is a non-contact (field) force.