Block (mass m) on smooth incline at angle θ. Axes: along slope (x), perpendicular to slope (y). Gravity resolved: mg sinθ along slope (down), mg cosθ into slope. Normal N perpendicular to slope (out). y-equation (no motion ⊥ to slope): N = mg cosθ. x-equation (motion along slope): ma = mg sinθ → a = g sinθ (mass independent, independent of friction on smooth surfaces). At θ = 90°: a = g (free fall). At θ = 30°: a = g/2. At θ = 0°: a = 0 (flat surface).

• Normal reaction N = mg cosθ < mg (since cosθ < 1 for 0 < θ < 90°). The normal force on an incline is LESS than the weight — more so for steeper inclines. At θ = 90° (vertical wall), N = 0. This means the effective weight pressing on the incline surface is mg cosθ, which determines friction force when friction is present (f = μmg cosθ).
• Mass independence: both N = mg cosθ and a = g sinθ are proportional to m in the numerator and m in denominator — mass cancels. Two blocks of different masses on the same smooth incline have the same acceleration. This is analogous to free fall (Galileo's principle) — gravity accelerates all masses equally. NEET trap: students sometimes divide by mass correctly but forget that N still depends on m (N = mg cosθ, not independent of mass).
• Key NEET kinematics: block starts from rest, slides down smooth incline of length l. a = g sinθ. Time: l = ½at² → t = √(2l/a) = √(2l/g sinθ). Velocity at bottom: v² = 2al → v = √(2al) = √(2gl sinθ) = √(2gh) where h = l sinθ. The velocity at the bottom depends only on the height h, independent of the slope angle.

When incline accelerates horizontally at b (to the right): block experiences gravity (mg downward) and normal force (N perpendicular to incline surface). In the lab frame, the block has unknown acceleration. Resolve all forces along and perpendicular to incline. Perpendicular: N − mg cosθ − mb sinθ = 0 (the horizontal acceleration of the incline has a component mb sinθ pressing into the slope and mb cosθ along the slope). So N = mg cosθ + mb sinθ. Along slope: mg sinθ − mb cosθ = ma → a = g sinθ − b cosθ (relative to incline surface — if positive, block slides down).

• Equilibrium condition (block stationary relative to incline): a = g sinθ − b cosθ = 0 → b = g tanθ. If the incline's horizontal acceleration equals g tanθ, the block appears stationary on the incline. This is because in the incline's (non-inertial) frame, the pseudo force exactly cancels the net component of gravity along the slope.
• If b < g tanθ: block slides DOWN the incline (a > 0 down slope). If b > g tanθ: block slides UP the incline (a < 0, meaning upward along slope). If b = 0 (incline at rest): a = g sinθ (standard result). Normal force always increases with b (N = mg cosθ + mb sinθ > mg cosθ).
• Alternative approach using pseudo force (non-inertial frame of incline): add pseudo force mb horizontally backward (opposite to incline's acceleration direction) on the block. Now solve as if the incline is at rest but with two 'gravity-like' forces: mg downward and mb backward. Resolve both along and perpendicular to incline. Result is identical. This approach is faster in NEET if you're comfortable with pseudo forces.

Combined system: one mass m₁ on incline (angle θ), connected via massless string over frictionless pulley to hanging mass m₂. Smooth incline: apply F_net = (m₁ + m₂)a to system. Driving force = m₂g − m₁g sinθ (if m₂ falls and m₁ goes up incline). a = (m₂ − m₁ sinθ)g / (m₁ + m₂). Tension T = m₁(g sinθ + a). For rough incline: add friction force f = μm₁g cosθ opposing relative motion of m₁ on incline.

• For the pulley-incline system to be in equilibrium (a = 0): m₂g = m₁g sinθ → m₂ = m₁ sinθ. The hanging mass equals the m₁ sinθ for the rough inclined version: m₂g = m₁g sinθ ± μm₁g cosθ (the ± accounts for which way the system tends to move). For m₁ to slide up: m₂g = m₁g sinθ + μm₁g cosθ → m₂ = m₁(sinθ + μ cosθ). For m₁ to slide down: m₂ = m₁(sinθ − μ cosθ). Equilibrium range: m₁(sinθ − μ cosθ) ≤ m₂ ≤ m₁(sinθ + μ cosθ).
• Rough incline alone (no pulley, block given initial velocity UP the slope): friction acts downward (opposing upward motion). Net deceleration = g sinθ + μg cosθ. After stopping, if block slides back: friction acts upward (now opposing downward motion). Acceleration down = g sinθ − μg cosθ. Condition for NOT sliding back: g sinθ ≤ μg cosθ → tanθ ≤ μ. If tanθ > μ: block slides back after stopping.
• NEET quick formula: for a rough incline, the angle of friction λ = tan⁻¹(μ). If θ > λ: block slides down on own. If θ < λ: block stays stationary even without applied force. If θ = λ: block is at the edge of sliding (limiting equilibrium).