1. Work Done by Constant & Variable Forces

Work done by constant force F causing displacement d: W = F · d = F d cos θ. Work is a scalar quantity (SI unit: Joule, J = N·m). Work is positive if 0° ≤ θ < 90°; zero if θ = 90° (e.g., normal reaction or centripetal force doing zero work); negative if 90° < θ ≤ 180° (e.g., kinetic friction). Work done by variable force: W = ∫ F(x) dx (graphically represented by the area under the Force-Displacement F-x curve).

2. Kinetic Energy & The Work-Energy Theorem

Kinetic Energy of particle of mass m moving with speed v: K = (1/2) m v^2 = p^2 / (2m), where p = mv is linear momentum. Work-Energy Theorem: The net work done by all forces (conservative, non-conservative, and external) acting on a body equals the change in its kinetic energy: W_net = ΔK = K_final - K_initial = (1/2)m(v² - u²). This theorem holds universally in inertial and non-inertial frames (incorporating pseudo-force work).

3. Potential Energy & Conservative Forces

For conservative forces, work done depends only on initial and final positions, independent of path. Relation to potential energy U: F = -dU/dx (or F = -∇U). Gravitational Potential Energy near earth's surface: U = mgh. Elastic Potential Energy stored in compressed/extended ideal spring (Hooke's law F = -kx): U = (1/2) k x^2. Conservation of Mechanical Energy: In presence of conservative forces only: K_i + U_i = K_f + U_f = constant.

4. Power & Mechanical Efficiency

Instantaneous Power: P = dW/dt = F · (dr/dt) = F · v = F v cos θ. Average Power: P_avg = Total Work / Total Time = ΔW / Δt. SI unit: Watt (W = J/s); Commercial unit: 1 Horsepower (1 hp) = 746 Watts; 1 Kilowatt-hour (kWh) = 3.6 × 10^6 Joules. Mechanical Efficiency of a machine: η = (Useful Work Output / Total Energy Input) × 100 = (Output Power / Input Power) × 100.

5. Uniform Circular Motion: Kinematics & Centripetal Dynamics

Angular displacement θ, angular velocity ω = dθ/dt = 2π/T = 2πf. Linear and angular relation: v = r ω. Centripetal (radial) acceleration: directed towards center: a_c = v² / r = ω² r. Centripetal force required: F_c = m v² / r = m ω² r. Tangential acceleration (changing speed): a_t = dv/dt = r α. Net acceleration: a_net = √(a_c² + a_t²). Skidding of vehicle on level circular road: Safe maximum speed v_max = √(μ s g r).

6. Banking of Roads & Vertical Circular Motion

Optimum speed on friction-free banked road of angle θ: v_0 = √(r g tan θ). With road friction μ: Maximum safe speed: v_max = √[r g (μ + tan θ) / (1 - μ tan θ)]; Minimum speed to prevent downward slip: v_min = √[r g (tan θ - μ) / (1 + μ tan θ)]. Vertical Circular Motion (string of length r, mass m): Critical speeds: (1) Minimum speed at lowest point to complete full loop: v_low = √(5gr); (2) Minimum speed at highest point: v_top = √(gr); (3) At horizontal position: v_mid = √(3gr). Tension difference between bottom and top is CONSTANT: T_bottom - T_top = 6mg.