MechanicsHigh Yield (2 Qs in JEE)

Laws of Motion & Friction Formulas

Newton's laws of motion, free body diagrams, static and kinetic friction, banking of roads, and pseudo-force.

7 Formulas•348+ Questions in Bank

Core Formulas & Mathematical Identities

Newton's Second Law & Momentum

\vec{F}_{net} = \frac{d\vec{p}}{dt} = m\vec{a}, \quad \vec{J} = \int \vec{F} \, dt = \Delta \vec{p}

Net force equals rate of change of linear momentum; Impulse J equals change in momentum.

Examiner Tip: Area under Force vs Time (F-t) graph gives impulse.

Static & Kinetic Friction

f_s \le \mu_s N, \quad f_k = \mu_k N \quad (\mu_s > \mu_k)

Limiting static friction and constant sliding kinetic friction.

Examiner Tip: Angle of repose θ equals angle of friction: tan θ = μ_s.

Banking of Roads — v_max, v_min & Optimum Speed

v_{\text{opt}} = \sqrt{rg\tan\theta}, \quad v_{\text{max}} = \sqrt{rg\cdot\frac{\tan\theta + \mu_s}{1 - \mu_s\tan\theta}}, \quad v_{\text{min}} = \sqrt{rg\cdot\frac{\tan\theta - \mu_s}{1 + \mu_s\tan\theta}}

On a banked road of radius r and bank angle θ: at optimum speed v_opt lateral friction is zero. Between v_min and v_max the car negotiates the bend safely. Below v_min, car slides inward (friction acts up the bank); above v_max, car skids outward (friction acts down the bank).

Examiner Tip: v_min exists only when tan θ > μ_s. For tan θ ≤ μ_s, even at rest the car won't slide inward. On unbanked road (θ = 0): v_max = √(μ_s rg) and v_min = 0.

Connected Bodies & Pulley Systems

a = \frac{\text{Net driving force}}{\text{Total mass}}, \quad T = \frac{2 m_1 m_2}{m_1 + m_2} g \; (\text{Atwood machine})

Standard acceleration and string tension formulas.

Examiner Tip: Tension in light inextensible string is uniform throughout unless passing over massive pulley.

Two-Block System & Slipping Threshold

a_{\text{common}} = \frac{F}{m_1 + m_2}, \quad F_{\text{max}} = \mu m_1 g \left(1 + \frac{m_2}{m_1}\right) = \mu(m_1 + m_2)g

Maximum external pulling force F applied on bottom block before top block m₁ starts to slip relative to m₂.

Examiner Tip: If applied force F ≤ F_max, both blocks accelerate together as a single combined unit.

Pseudo-Force & Apparent Weight in Elevator

\vec{F}_{\text{pseudo}} = -m \vec{a}_{\text{frame}}, \quad N = m(g \pm a) \quad (+ \text{ for upward } a, \; - \text{ for downward } a)

Inertial force applied to masses when observed from a non-inertial frame accelerating at a_frame.

Examiner Tip: In free fall (a = g downwards), apparent weight N = 0 (weightlessness).

Virtual Work & String Constraint Relations

\sum \vec{T} \cdot \vec{x} = 0 \implies \sum \vec{T} \cdot \vec{v} = 0 \implies \sum \vec{T} \cdot \vec{a} = 0

Total virtual work done by tension in an inextensible, massless string system is identically zero.

Examiner Tip: Wedge constraint: Relative velocity components perpendicular to the common inclined contact face must be equal.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Laws of Motion & Friction

•Static friction is SELF-ADJUSTING up to limiting friction f_s ≤ μ_s N (it is not always equal to μ_s N).
•Pseudo force applies ONLY in non-inertial (accelerating) reference frames: F_pseudo = -m a_frame.

Fundamental Theory & Core Concepts

4 Topics

1Newton's Three Laws of Motion & Momentum

›First Law (Law of Inertia): A body continues in its state of rest or uniform motion in a straight line unless acted upon by a non-zero external net force. Defines inertial reference frames.
›Second Law: The time rate of change of linear momentum of a body is directly proportional to the applied net external force and occurs along the direction of that force: F_net = dp/dt = m(dv/dt) + v(dm/dt). For constant mass, F_net = ma.
›Third Law: For every action force, there is an equal and opposite reaction force (F_AB = -F_BA). Action and reaction forces act simultaneously on DIFFERENT interacting bodies, so they never cancel each other.

2Friction Mechanics: Static, Limiting & Kinetic

›Origin: Friction arises from molecular electrostatic adhesion and microscopic interlocking of surface irregularities between two bodies in contact.
›Static Friction (f_s): Self-adjusting tangential force that prevents relative motion. 0 ≤ f_s ≤ f_s(max) = μ_s N. If applied force F < μ_s N, then f_s = F exactly.
›Kinetic Friction (f_k): Opposes ongoing relative sliding motion between surfaces: f_k = μ_k N. Nearly independent of relative velocity and apparent contact area (μ_s > μ_k > μ_rolling).
›Angle of Friction (λ) and Repose (θ_r): tan λ = μ_s. The angle of repose θ_r is the minimum inclination of an inclined plane at which a body placed on it begins to slide down under gravity: tan θ_r = μ_s.

3Reference Frames & Pseudo Forces

›Inertial Frame: A frame of reference that is either at rest or moving with constant velocity (zero acceleration) relative to distant stars. Newton's laws hold directly.
›Non-Inertial Frame: An accelerating or rotating frame of reference (acceleration a_0). Newton's laws do not hold directly.
›Pseudo Force: An apparent fictitious force introduced to apply Newton's second law in an accelerating frame: F_pseudo = -m a_0, acting through the center of mass in the direction exactly OPPOSITE to the frame's acceleration.

4Circular Motion Dynamics & Road Banking

›Centripetal Force: The net inward radial force required to maintain circular motion of radius r at speed v: F_c = m v² / r = m ω² r. It is NOT a separate physical force; it is provided by existing forces (tension, friction, normal, gravity).
›Level Curved Road: Friction alone provides centripetal force: v_max = √(μ_s g r).
›Banked Road: Road surface is inclined at angle θ with horizontal. Optimum speed without friction: v_opt = √(g r tan θ). Range with friction: v_min = √[g r (tan θ - μ_s) / (1 + μ_s tan θ)] and v_max = √[g r (tan θ + μ_s) / (1 - μ_s tan θ)].

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