Newton's Laws of Motion
Complete Formula Sheet & Shortcut Bible · BITSAT 2026
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Newton's Core Laws & Momentum
Newton's First Law
ΣF = 0 ⇔ v = constant
Law of Inertia. An object stays at rest or in uniform motion unless a net external force acts on it.
Newton's Second Law
Fnet = ma
Fnet = dp/dt
Fnet = dp/dt
The fundamental law connecting force, mass, and acceleration. The rate of change of momentum is force.
Newton's Third Law
F₁₂ = -F₂₁
For every action, there is an equal and opposite reaction. Forces occur in pairs on different bodies.
Linear Momentum (p)
p = mv
A vector quantity. Its direction is the same as the velocity's direction.
Impulse (J)
J = Favg ⋅ Δt = Δp
Impulse is the change in momentum. It's the area under the Force-Time graph.
Inclined Plane Analysis
Force Components on Inclined Plane
Force parallel: mg sinθ
Force perpendicular: mg cosθ
Force perpendicular: mg cosθ
Always resolve the weight 'mg' into components parallel and perpendicular to the incline.
Acceleration (Smooth Incline)
a = g sinθ
Acceleration is independent of the mass of the object. It only depends on the angle of inclination.
Acceleration (Rough Incline)
Sliding down: a = g(sinθ - μₖcosθ)
Pushed up: a = g(sinθ + μₖcosθ)
Pushed up: a = g(sinθ + μₖcosθ)
Friction (fₖ = μₖN = μₖmgcosθ) always opposes the direction of motion or intended motion.
Normal Force on Incline
N = mg cosθ
Normal force is NOT always equal to mg. It balances the perpendicular component of weight.
Pulley Systems (Atwood Machine)
Acceleration (Simple Atwood)
a = (m₂ - m₁)g
(m₂ + m₁)
(m₂ + m₁)
For m₂ > m₁. The net driving force is the difference in weights.
Tension (Simple Atwood)
T = 2m₁m₂g
(m₁ + m₂)
(m₁ + m₂)
This is the harmonic mean of the weights. Tension is the same throughout the massless string.
Modified Atwood (Block on Table)
Smooth Table: a = m₂g
(m₁ + m₂)
Rough Table: a =(m₂ - μₖm₁)g
(m₁ + m₂)
(m₁ + m₂)
Rough Table: a =
(m₁ + m₂)
The driving force is the hanging weight, opposed by friction on the table block if present.
BITSAT Pulley & Lift Hacks
Global Formula: For any system, Acceleration = (Net Driving Force) / (Total Mass being Moved). Avoids simultaneous equations.
Tension Check: In a simple Atwood machine, tension T is always between m₁g and m₂g. Use this to eliminate options quickly.
'g-effective' for Lifts: When a lift accelerates, use g_eff. Upwards: g_eff = g+a. Downwards: g_eff = g-a. Then use standard formulas.
Constraint Motion: If one part of a string moves by 'x', find how much the other masses move. The ratio of displacements gives the ratio of accelerations.
Circular Motion Dynamics
Centripetal Acceleration (a꜀)
a꜀ = v²
r = ω²r
r = ω²r
Always directed towards the center of the circle. It changes the direction of velocity, not its magnitude.
Centripetal Force (F꜀)
F꜀ = ma꜀ = mv²
r
r
Not a new force. It's the net force towards the center (e.g., Tension, Gravity, Friction).
Banking of Roads (No Friction)
tanθ = v²
rg
rg
The horizontal component of the Normal force provides the required centripetal force.
Max Speed on Banked Road (with Friction)
v²max = rg (μₛ + tanθ)
(1 - μₛtanθ)
(1 - μₛtanθ)
Both Normal force component and friction contribute to the centripetal force.
Conical Pendulum
tanθ = v²
rg
T = mg/cosθ
rg
T = mg/cosθ
The horizontal component of tension (Tsinθ) provides the centripetal force.
Frames of Reference: Key Differences
| Property | Inertial Frame | Non-Inertial Frame |
|---|---|---|
Definition | Frame at rest or moving with constant velocity. | Frame that is accelerating. |
Newton's 1st Law | Holds true. An object with no net force remains at rest or in uniform motion. | Does not hold true without introducing a pseudo force. |
Forces Required | Only REAL forces (gravity, tension, friction, etc.) are considered. | REAL forces + a PSEUDO force (Fₚ = -ma₀) must be applied to use F=ma. |
BITSAT Example | A train moving at a constant 90 km/h. The Earth (approximated). | An accelerating elevator. A car turning a corner. A rotating merry-go-round. |