Rotational Dynamics
Rotational Dynamics is one of the heaviest numerical chapters in Maharashtra HSC Physics. The paper tests moment of inertia of standard bodies, the parallel and perpendicular axes theorems, rolling motion, banking of roads and vertical circular motion, usually as one derivation plus one numerical.
Core rules
- Moment of inertia I = Σmr²; radius of gyration k is defined by I = Mk².
- Parallel axes: I = I(cm) + Md². Perpendicular axes (laminae only): I(z) = I(x) + I(y).
- Standard values: ring about a diameter MR²/2, disc about a diameter MR²/4, solid sphere 2MR²/5, hollow sphere 2MR²/3, thin rod about its centre ML²/12.
- Rotational analogues: τ = Iα, L = Iω, rotational KE = ½Iω², power = τω.
- Rolling without slipping: v = Rω, a = Rα, total KE = ½Mv²(1 + k²/R²), and on an incline a = g sinθ/(1 + k²/R²).
- Frictionless banking: v = √(rg tanθ). Vertical circle: minimum speed at the top √(gr), at the bottom √(5gr).
Worked example
A solid disc (M = 2 kg, R = 0.5 m) rolls down a 30° incline without slipping. For a disc k²/R² = ½, so a = g sin30°/(1 + 0.5) = (9.8 × 0.5)/1.5 = 3.27 m/s². Static friction acts up the incline, yet it does no work because the contact point is instantaneously at rest; mechanical energy is therefore conserved.
Traps
- Using I = MR² for a disc instead of ½MR².
- Believing angular momentum is conserved whenever net force is zero; it needs zero net external torque.
- Confusing v = Rω (rolling or fixed-axis rotation) with the general velocity of a point on a rotating body.