MechanicsTop Priority (3-4 Qs in JEE)

Rotational Motion & Rigid Bodies Formulas

Moment of inertia, torque, angular momentum conservation, rolling motion without slipping, and rotational kinetic energy.

8 Formulas•665+ Questions in Bank•

Concept Diagram

Uniform Circular Motion & Centripetal Acceleration

Circular path with radius r, tangential velocity v, and inward radial acceleration ac = v²/r.

Core Formulas & Mathematical Identities

Moment of Inertia of Standard Bodies

I_{\text{ring}} = MR^2, \; I_{\text{disc}} = \frac{1}{2}MR^2, \; I_{\text{solid sphere}} = \frac{2}{5}MR^2, \; I_{\text{rod}} = \frac{1}{12}ML^2

Moments of inertia about central symmetry axes.

Examiner Tip: Hollow sphere has I = (2/3)MR².

Parallel & Perpendicular Axis Theorems

I = I_{cm} + Md^2 \; (\text{Parallel}), \quad I_z = I_x + I_y \; (\text{Perpendicular, 2D laminas})

Translating and rotating axes of inertia.

Examiner Tip: Parallel axis theorem requires I_{cm} through center of mass, NOT any random point.

Torque & Angular Momentum

\vec{\tau} = \vec{r} \times \vec{F} = I\vec{\alpha}, \quad \vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}, \quad \vec{\tau}_{ext} = \frac{d\vec{L}}{dt}

Rotational analogs of Newton's second law.

Examiner Tip: If net external torque is zero, angular momentum is conserved (I₁ω₁ = I₂ω₂).

Pure Rolling Motion Kinetic Energy

K_{total} = K_{trans} + K_{rot} = \frac{1}{2}Mv_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)

Total kinetic energy of body rolling without slipping with radius of gyration k.

Examiner Tip: For pure rolling: v_{cm} = Rω and a_{cm} = Rα.

Acceleration on an Inclined Plane

a = \frac{g\sin\theta}{1 + \frac{k^2}{R^2}}, \quad \mu_{min} = \frac{\tan\theta}{1 + \frac{R^2}{k^2}}

Linear acceleration of a rolling body down an incline of angle θ.

Examiner Tip: Body with smallest k²/R² reaches the bottom first (Solid Sphere > Disc > Hollow Sphere > Ring).

Angular Momentum in Combined Motion (Translation + Rotation)

\vec{L}_O = \vec{r}_{cm} \times M\vec{v}_{cm} + I_{cm}\vec{\omega}, \quad K = \frac{1}{2}I_{\text{IAOR}}\,\omega^2

Total angular momentum about origin O decomposes into orbital CM term plus spin intrinsic term.

Examiner Tip: Pure rolling can be treated as pure rotation about the Instantaneous Axis of Rotation (IAOR) passing through the contact point.

Rolling Kinematics & Point Velocity on Wheel

v_P = 2 v_{cm} \sin\left(\frac{\theta}{2}\right) \implies v_{\text{top}} = 2 v_{cm}, \quad v_{\text{bottom}} = 0, \quad a_{\text{contact}} = \frac{v_{cm}^2}{R} \; (\text{towards center})

Speed of any point on perimeter of rolling cylinder at angle θ from bottom contact point.

Examiner Tip: The bottom contact point is instantaneously at rest (v = 0), but its instantaneous acceleration is non-zero (directed toward the center).

Toppling vs Sliding of a Block

\text{Toppling before sliding condition: } \mu > \frac{b}{h}, \quad F_{\text{topple}} = \frac{mg b}{2h}

Critical threshold for block of base width b and height h pushed by horizontal force at top edge.

Examiner Tip: Normal reaction shifts forward by distance x = (F·h)/(mg) as horizontal force F increases, reaching the front corner (x = b/2) at the verge of toppling.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Rotational Motion & Rigid Bodies

•Applying perpendicular axis theorem to 3D solid bodies (it is valid ONLY for planar 2D laminar objects).
•Forgetting that friction does NO work in pure rolling without slipping.

Previous Chapter

Center of Mass & Momentum Conservation

7 Formulas•Mechanics

Next Chapter

Gravitation & Satellite Motion

Mechanics•8 Formulas