Chapter 7 — Core Concepts
Building the Foundation
Five big ideas that explain how rigid bodies move
NCERT §7.1 · Rigid Body
What is a Rigid Body?
Imagine holding a steel rod. You can throw it, spin it, push it — but the distance between any two tiny particles inside it never changes. That is a rigid body.
Definition
A rigid body is an object in which the distance between every pair of particles stays constant, no matter what forces act on it.
Ideal vs Real
- A perfectly rigid body is an idealization — exists only in theory.
- Real objects like rubber balls deform slightly — but for Physics problems, we treat them as rigid.
- Cricket bat, spinning top, wheel — all treated as rigid bodies.
Exam Tip
"Rigid body" in a question → deformation is zero. The body keeps its shape perfectly.
NCERT §7.1 · Translational Motion
Translational Motion
Slide a book across a table. Every single part of the book moves in the same direction, covering the same distance at the same time. No spinning. That is pure translation.
Key Idea
In pure translational motion — every particle has the same velocity at any instant.
Two sub-types
- Rectilinear — straight line path. Example: car on a highway.
- Curvilinear — curved path. Example: cricket ball thrown at an angle.
In translational motion, you only need to track one point — the Centre of Mass — to describe the whole body's motion.
NCERT §7.1 · Rotational Motion
Rotational Motion
Now imagine a spinning wheel. The axle stays fixed. Every point on the wheel moves in a circle around the axle. Every particle traces a circular path — but those farther out move faster.
In pure rotation — every particle has the same angular velocity ω, but different linear speeds (v = rω, so bigger r → bigger v).
Common confusion: Angular velocity ω is the SAME for all particles. But linear speed v = rω — so outer particles move faster!
NCERT §7.12 · Rolling Motion
Rolling Motion = Translation + Rotation
A ball rolling on the ground is doing both at the same time: its centre moves forward (translation) while it spins (rotation).
Translation
v_cm
+
v_cm
Rotation
ω about CM
=
ω about CM
Pure Rolling
v_cm = Rω
v_cm = Rω
Speeds at key points
- Contact point (bottom): v = 0 — no slipping
- Centre of mass: v = v_cm
- Topmost point: v = 2 × v_cm
Shortcut: From bottom to top — multiply v_cm by 0, 1, 2. Contact = 0, Centre = v_cm, Top = 2v_cm.
NCERT §7.1 · Centre of Mass
Centre of Mass (COM)
Balance a cricket bat on your finger. There is one special point where it balances perfectly. That is very close to the Centre of Mass — the "average position" of all the mass.
The Centre of Mass is the mass-weighted average position of all particles. The entire mass can be assumed concentrated here for describing translational motion.
x_cm = (m₁x₁ + m₂x₂) / (m₁ + m₂)
For two particles · General form: x_cm = Σ(mᵢxᵢ) / Σmᵢ · COM lies between them, closer to the heavier mass
Symmetric bodies — instant shortcuts
- Uniform rod → midpoint
- Circle / disk / sphere → geometric centre
- Square, rectangle → intersection of diagonals
Watch out! COM does NOT have to be inside the body. For a ring, COM is at the centre — where there is no material at all!
NCERT §7.2 · Motion of Centre of Mass
Motion of the Centre of Mass
This is the most powerful idea in the chapter — it simplifies everything.
The COM of a system moves as if the entire mass is concentrated there and all external forces act on it — exactly like a point particle.
F_ext = M × a_cm
F_ext = total external force on system · M = total mass · a_cm = acceleration of COM · Internal forces have no effect on COM
Why this matters
- A bat thrown in the air spins and tumbles — looks complex. Its COM traces a simple parabola.
- A bomb explodes mid-air — pieces scatter. The COM of all pieces continues on the same path.
- Internal forces cancel in pairs — they cannot shift the COM.
NEET Favourite
No external force → COM velocity is constant. Use this for explosion and "man on a boat" problems instantly.
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Angular Variables
θ — Angular Displacement
Angle rotated by body · Unit: radians (rad) · 1 revolution = 2π rad = 360°
ω = dθ/dt — Angular Velocity
Rate of change of angle · Unit: rad/s · Same for ALL particles of a rigid body · ω = 2πn (n = revolutions per second)
α = dω/dt — Angular Acceleration
Rate of change of angular velocity · Unit: rad/s² · Positive = speeding up · Negative = slowing down
Linear ↔ Angular Relations
v = r × ω
v = linear speed · r = distance from axis · ω = angular velocity · Farther from axis → faster linear speed
aₜ = r × α
Tangential acceleration — along the direction of motion along circle · r = radius from axis
aₙ = rω² = v²/r
Centripetal acceleration — directed towards the axis · Always present in circular/rotational motion
Rotational Equations of Motion
These are exact analogues of the linear kinematic equations. Replace s→θ, v→ω, u→ω₀, a→α.
| Linear Equation | Rotational Equivalent |
|---|---|
| v = u + at | ω = ω₀ + αt |
| s = ut + ½at² | θ = ω₀t + ½αt² |
| v² = u² + 2as | ω² = ω₀² + 2αθ |
Huge shortcut: You already know these — they are the same SUVAT equations with different symbols. No extra memorisation needed.
Centre of Mass Formulas
x_cm = Σ(mᵢxᵢ) / Σmᵢ
General COM for any number of particles · Apply same formula for y_cm and z_cm
v_cm = Σ(mᵢvᵢ) / Σmᵢ = p_total / M
Velocity of COM = Total momentum / Total mass
F_ext = M × a_cm
Newton's 2nd Law for a system · Only EXTERNAL forces accelerate the COM
Rolling Motion Formulas
v_cm = Rω · Pure Rolling Condition
R = radius · ω = angular velocity · Holds only when there is NO slipping
v_top = 2v_cm · v_bottom = 0
Topmost point: twice the speed of centre · Contact point: zero velocity (no-slip condition)
Side-by-Side Analysis
Comparisons & Contrasts
The clearest way to avoid mix-ups in exams
Translation vs Rotation — Full Table
| Property | Translational | Rotational |
|---|---|---|
| Position | Displacement s (m) | Angular disp. θ (rad) |
| Rate of change | Velocity v (m/s) | Angular vel. ω (rad/s) |
| Rate of rate | Acceleration a (m/s²) | Angular acc. α (rad/s²) |
| Cause of motion | Force F (N) | Torque τ (N·m) |
| Inertia | Mass m (kg) | Moment of Inertia I (kg·m²) |
| Newton's 2nd | F = ma | τ = Iα |
| Kinetic Energy | ½mv² | ½Iω² |
| Momentum | p = mv | L = Iω |
| Equation 1 | v = u + at | ω = ω₀ + αt |
| Equation 2 | s = ut + ½at² | θ = ω₀t + ½αt² |
| Equation 3 | v² = u² + 2as | ω² = ω₀² + 2αθ |
Rotational Motion vs Circular Motion
Very common confusion: Students mix up "rotational motion of a body" with "circular motion of a particle". They describe different things.
Rotational Motion
• Describes a rigid body
• The whole body rotates about axis
• Example: spinning fan, top
• Axis can be fixed or moving
• The whole body rotates about axis
• Example: spinning fan, top
• Axis can be fixed or moving
Circular Motion
• Describes a single particle
• That one particle moves in circle
• Example: stone tied to a string
• Particle moves around a centre
• That one particle moves in circle
• Example: stone tied to a string
• Particle moves around a centre
A rotating fan: the fan has rotational motion. Each tip of the fan has circular motion. Same physical event — two different descriptions!
Internal vs External Forces
Internal Forces
• Between particles of the system
• Always come in Newton's 3rd law pairs
• Net sum = zero (cancel out)
• Do NOT affect COM
• Always come in Newton's 3rd law pairs
• Net sum = zero (cancel out)
• Do NOT affect COM
External Forces
• From outside the system
• Gravity, normal, friction, etc.
• Do not cancel automatically
• F_ext = M × a_cm
• Gravity, normal, friction, etc.
• Do not cancel automatically
• F_ext = M × a_cm
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