Class 11 · Chapter 4

Laws of Motion

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Laws of Motion Formula Sheet

15 formulas across 4 topics in Laws of Motion.

2 min read

Updated 2026-07-03 · v1.0.1

Newton's Laws & Momentum

Force is the rate of change of momentum. Every other result in this chapter is bookkeeping around this single idea.

F = dp/dt = ma

Newton's second law

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F = ma holds only when mass is constant; F = dp/dt is the general form

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F	net external force	N	[MLT⁻²]
p	linear momentum (mv)	kg·m/s	[MLT⁻¹]
m	mass	kg	[M]
a	acceleration	m/s²	[LT⁻²]

Valid when

F must be the NET force
Valid in inertial frames only

Common mistakes

Using F = ma for variable-mass systems (rocket, falling chain) — use F = dp/dt there
Forgetting that a is along the NET force, not along the motion

Read the concept →

J = F Δt = Δp

Impulse–momentum theorem

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Integrating F = dp/dt over the contact time

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
J	impulse	N·s	[MLT⁻¹]
Δp	change in momentum	kg·m/s	[MLT⁻¹]
Δt	duration of force	s	[T]

Valid when

For variable force, J = area under the F–t graph

Common mistakes

Impulse equals CHANGE in momentum (vector subtraction) — for a ball bouncing back, magnitudes add

Read the concept →

N = m(g ± a)

Apparent weight in a lift

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Newton's second law on the person: N − mg = ±ma

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
N	normal reaction (apparent weight)	N	[MLT⁻²]
a	lift's acceleration	m/s²	[LT⁻²]

Valid when

+a when the lift accelerates upward
−a when it accelerates downward
N = 0 in free fall (weightlessness)

Common mistakes

Direction of VELOCITY is irrelevant — only the direction of acceleration decides the sign

Read the concept →

F(pseudo) = −m a(frame)

Pseudo force (non-inertial frame)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
m	mass of the object being analysed	kg	[M]
a(frame)	acceleration of the reference frame	m/s²	[LT⁻²]

Valid when

Apply ONLY when solving from inside an accelerating frame
Direction is opposite to the frame's acceleration

Common mistakes

Adding a pseudo force while also working in the ground frame — double counting; pick one frame and stay in it

Read the concept →

Worth remembering

Newton's third law pairs act on DIFFERENT bodies — they never cancel each other
A body moving at constant velocity has zero net force on it, no matter how fast it moves

NEET tests the laws directly; JEE hides them inside connected-body and constraint problems — always start with a free body diagram.

Connected Bodies & Strings

Same string, same tension (if massless and frictionless pulley); same rigid connection, same acceleration magnitude.

a = (m₁ − m₂)g / (m₁ + m₂), T = 2m₁m₂g / (m₁ + m₂)

Atwood machine (two masses over a pulley)

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Newton's second law on each mass, tension common

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
m₁, m₂	the two hanging masses (m₁ > m₂)	kg	[M]
T	string tension	N	[MLT⁻²]

Valid when

Massless, inextensible string
Frictionless, massless pulley

Common mistakes

T is NOT the average of the two weights — it always lies between m₂g and m₁g, closer to the geometric mean

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a = F / (m₁ + m₂), contact force = m₂F / (m₁ + m₂)

Two blocks pushed together on a smooth surface

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System equation for a; second law on the rear block alone for contact force

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
F	applied force (on block m₁)	N	[MLT⁻²]
m₁, m₂	front and rear block masses	kg	[M]

Valid when

Smooth surface
Blocks stay in contact

Common mistakes

Contact force changes if you push from the other side — it becomes m₁F/(m₁+m₂)

Read the concept →

Worth remembering

For n identical connected blocks pulled by force F, tension after the kth block from the front = F(n−k)/n

Friction

Static friction is self-adjusting up to a maximum; kinetic friction is constant. Confusing the two is the #1 error in this chapter.

f(s,max) = μₛN

Maximum static friction (limiting friction)

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
μₛ	coefficient of static friction	dimensionless	[M⁰L⁰T⁰]
N	normal reaction	N	[MLT⁻²]

Valid when

Actual static friction f ≤ μₛN — it equals whatever is needed to prevent sliding, up to this cap

Common mistakes

Writing f = μₛN when the body is at rest with a small applied force — static friction equals the APPLIED force there, not its maximum

Read the concept →

f(k) = μₖN

Kinetic friction

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
μₖ	coefficient of kinetic frictionμₖ < μₛ for the same pair of surfaces	dimensionless	[M⁰L⁰T⁰]
N	normal reaction	N	[MLT⁻²]

Valid when

Surfaces sliding relative to each other
Direction opposes RELATIVE sliding

Common mistakes

N is not always mg — on an incline N = mg cosθ; with a pulling force at angle θ, N = mg − F sinθ

Read the concept →

tan θ(repose) = μₛ

Angle of repose

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On the verge of sliding on an incline: mg sinθ = μₛ mg cosθ

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ(repose)	maximum incline angle before sliding starts	rad (or °)	[M⁰L⁰T⁰]

Valid when

Body on the verge of sliding under gravity alone

Common mistakes

Angle of repose equals angle of friction (tan⁻¹ μₛ) — two names, one number

Read the concept →

a = g(sinθ − μₖ cosθ)

Acceleration sliding down a rough incline

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Second law along the incline with kinetic friction acting up the slope

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ	incline angle	rad (or °)	[M⁰L⁰T⁰]
μₖ	coefficient of kinetic friction	dimensionless	[M⁰L⁰T⁰]

Valid when

Sliding down; for a block pushed UP the incline, friction reverses: a = g(sinθ + μₖ cosθ) deceleration

Read the concept →

Worth remembering

Friction can be the FORWARD force — it is what accelerates a walking person and a car's driven wheels
Friction is independent of contact area (for the ideal dry-friction model used in JEE/NEET)

JEE Advanced favourite: blocks stacked on blocks, asking whether they move together or slip — compare required friction with maximum available friction.

Dynamics of Circular Motion

Centripetal force is not a new force — it is the NET inward force required to keep the body on the circle. Something real (tension, friction, gravity, normal) must supply it.

F(c) = mv²/r = mω²r

Centripetal force

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	speed along the circle	m/s	[LT⁻¹]
r	radius of the circle	m	[L]
ω	angular speed	rad/s	[T⁻¹]

Valid when

Directed toward the centre, always perpendicular to velocity

Common mistakes

Adding centripetal force as an extra force in the free body diagram — it is the RESULT of real forces, not one of them

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v(max) = √(μₛ g r)

Maximum speed on a flat (unbanked) turn

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Friction supplies the centripetal force: μₛmg = mv²/r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
μₛ	coefficient of static friction (tyres–road)	dimensionless	[M⁰L⁰T⁰]
r	radius of the turn	m	[L]

Valid when

Flat road
Static friction (tyres roll without skidding)

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tan θ = v²/rg

Ideal banking angle (no friction needed)

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Horizontal component of normal reaction supplies the centripetal force

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
θ	banking angle	rad (or °)	[M⁰L⁰T⁰]
v	design speed of the turn	m/s	[LT⁻¹]

Valid when

At exactly this speed, no friction is required

Common mistakes

With friction included, there is a RANGE of safe speeds around the design speed — JEE asks for v(max) with friction: v² = rg(tanθ + μ)/(1 − μ tanθ)

Read the concept →

v(top,min) = √(gr), v(bottom,min) = √(5gr)

Minimum speeds in a vertical circle (string)

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At the top, gravity alone supplies centripetal force (T = 0); energy conservation connects top to bottom

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
r	radius of the vertical circle	m	[L]

Valid when

String or track that can only PULL/PUSH inward — for a rod, v(top,min) = 0

Common mistakes

Applying √(5gr) to a rod or a tube — the 5gr result is only for strings where tension cannot be negative

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T(period) = 2π √(L cosθ / g); tanθ = v²/rg

Conical pendulum

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Vertical balance T cosθ = mg with horizontal centripetal equation T sinθ = mv²/r

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
L	string length	m	[L]
θ	half-angle of the cone (string from vertical)	rad (or °)	[M⁰L⁰T⁰]
r	radius of the horizontal circle (= L sinθ)	m	[L]

Valid when

The bob can never swing fully horizontal — θ → 90° needs infinite speed
Period depends on L cosθ (the vertical height), not on L alone

Common mistakes

Tension here EXCEEDS mg (T = mg/cosθ) — unlike a hanging pendulum at rest

Read the concept →

Worth remembering

Tension in a vertical circle is maximum at the bottom and minimum at the top: T(bottom) − T(top) = 6mg for a string
In uniform circular motion, speed is constant but velocity is not — there is always acceleration

NEET loves the √(5gr) one-liner; JEE Advanced replaces the string with rods, tubes, and inside-surface tracks — re-derive from the free body diagram each time.

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