Laws of Motion — Complete Formula

Newton’s Laws of Motion

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Newton’s Second Law	[\vec{F} = m\vec{a}]	F = force, m = mass, a = acceleration	N	Most important law ⭐
Momentum Form	[\vec{F} = \dfrac{d\vec{p}}{dt}]	p = momentum, t = time	N	Used when mass/velocity changes
Newton’s Third Law	[\vec{F}_{AB} = -\vec{F}_{BA}]	[F_{AB}] = force by A on B, [F_{BA}] = force by B on A	N	Equal & opposite forces ⭐

💡 Memory Hint:
Every problem → reduce to [F = ma]

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Linear Momentum	[\vec{p} = m\vec{v}]	p = momentum, m = mass, v = velocity	kg·m/s	Vector quantity
Impulse	[\vec{J} = \vec{F} \cdot t]	J = impulse, F = force, t = time	N·s	Force applied over time ⭐
Impulse-Momentum	[\vec{F}t = \Delta \vec{p}]	Δp = change in momentum	—	Used in collisions ⭐

💡 Memory Hint:
Impulse = change in momentum

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Static Friction	[f_s \leq \mu_s N]	[f_s] = static friction, [μ_s] = coefficient, N = normal reaction	N	Self-adjusting ⚠️
Limiting Friction	[f_{max} = \mu_s N]	[f_{max}] = maximum static friction	N	Just before motion ⭐
Kinetic Friction	[f_k = \mu_k N]	[f_k] = kinetic friction, [μ_k] = coefficient	N	Constant during motion
Angle of Friction	[\tan\theta = \mu]	θ = angle of friction, μ = coefficient	—	Useful in incline problems
Angle of Repose	[\tan\theta = \mu]	θ = angle of repose	—	Same as friction angle ⭐

💡 Memory Hint:
Static → varies
Kinetic → fixed

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Weight Components	[mg\sin\theta,\quad mg\cos\theta]	m = mass, g = gravity, θ = angle of incline	N	Resolve along plane ⭐
Net Force	[F = mg\sin\theta – f]	f = friction force	N	Check direction ⚠️
Acceleration (no friction)	[a = g\sin\theta]	a = acceleration	m/s²	Basic result ⭐
Acceleration (with friction)	[a = g(\sin\theta – \mu\cos\theta)]	μ = coefficient of friction	m/s²	Very important ⭐

💡 Memory Hint:
Along slope → sinθ
Perpendicular → cosθ

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
General Equation	[F = ma]	F = net force, m = mass, a = acceleration	—	Apply separately to each body ⭐
Acceleration (Atwood Machine)	[a = \dfrac{(m_1 – m_2)g}{m_1 + m_2}]	m₁, m₂ = masses, g = gravity	m/s²	Standard result ⭐
Tension (Atwood)	[T = \dfrac{2m_1 m_2 g}{m_1 + m_2}]	T = tension	N	Frequently asked ⭐

💡 Memory Hint:
Heavier mass → moves down

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Centripetal Force	[F = \dfrac{mv^2}{r}]	m = mass, v = velocity, r = radius	N	Towards center ⭐
Acceleration	[a = \dfrac{v^2}{r}]	a = centripetal acceleration	m/s²	Direction changes ⚠️
Angular Form	[F = m\omega^2 r]	ω = angular velocity	N	Alternative form

💡 Memory Hint:
Circular motion = constant speed, changing direction

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Pseudo Force	[F_{pseudo} = -m a_{frame}]	[a_{frame}] = acceleration of frame	N	Opposite direction ⭐
Effective Gravity	[g_{eff} = g \pm a]	g = gravity, a = frame acceleration	m/s²	Lift problems ⭐

💡 Memory Hint:
Accelerating frame → add pseudo force

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