System of Particles and Rotational Motion

A rigid body is an idealised body whose shape and size do not change under an applied force — the distance between any two particles stays fixed. Real bodies deform slightly, but for mechanics we treat solids as rigid.

The centre of mass (COM) of a system of particles is the point at which the entire mass can be imagined concentrated for describing overall translational motion. For two particles on the x-axis at x₁, x₂: X = (m₁x₁ + m₂x₂)/(m₁ + m₂). Equal masses give a midpoint COM; a heavier mass pulls the COM towards itself.

For n particles in 3-D: X = (Σm_ix_i)/M, and similarly for Y, Z, where M = Σm_i. In vector form, R = (Σm_ir_i)/M; for a continuous body, R = (1/M)∫r dm.

By symmetry, the COM of a uniform ring, disc, sphere or cube lies at its geometric centre — for a ring, this is the empty space at the centre, not on the ring material itself.

Differentiating R = (Σm_ir_i)/M with respect to time gives the velocity of the COM: MV = Σm_iv_i = total linear momentum p of the system. So the total momentum of a system of particles equals the total mass times the velocity of its COM — exactly as if all the mass were concentrated at the COM and moving with velocity V.

Differentiating again, MA = Σm_ia_i = F_ext, the vector sum of all external forces (internal forces between particles cancel in pairs by Newton's third law and do not affect COM motion). This is the key result: the centre of mass of a system moves as if the entire mass of the system were concentrated there and all the external forces acted directly at that point.

This is why a projectile that explodes in mid-air under purely internal forces (the explosion) has its COM continuing along the same parabolic path as before the explosion — the fragments fly off in different directions, but their combined COM does not change its trajectory because no new external force has appeared.

Combining the results above, dp/dt = MA = F_ext. This is Newton's second law extended to a system of particles: the rate of change of total linear momentum of a system equals the total external force on it.

If F_ext = 0, then p = MV is constant — this is the law of conservation of linear momentum for a system, and it also means the velocity of the COM stays constant when no external force acts, even if the particles that make up the system keep colliding, exploding or rearranging among themselves.

This principle underlies recoil of a gun, motion of a rocket, and collision problems: as long as no external force acts along a direction, momentum along that direction is conserved for the system as a whole, whatever happens internally.

Torque and angular momentum are both defined using the vector (cross) product. For two vectors a and b, a × b = ab sinθ n̂, where θ is the angle between them and n̂ is a unit vector perpendicular to the plane of a and b, given by the right-hand rule: curl the fingers from a to b, and the thumb points along n̂.

Key properties: (i) a × b = −(b × a) — not commutative; reversing the order reverses the direction. (ii) a × a = 0, since sinθ = 0, so parallel or antiparallel vectors give a × b = 0. (iii) |a × b| equals the area of the parallelogram formed by a and b. (iv) In components, a × b = (a_yb_z − a_zb_y)î + (a_zb_x − a_xb_z)ĵ + (a_xb_y − a_yb_x)k̂.

For unit vectors: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ, and î × î = ĵ × ĵ = k̂ × k̂ = 0.

For a rigid body rotating about a fixed axis, every particle moves in a circle centred on the axis, and all particles sweep the same angle in the same time — so a single angular velocity ω describes the whole body, even though different particles have different linear speeds depending on their distance from the axis.

Angular velocity is a vector directed along the axis of rotation (right-hand rule: curl fingers in the sense of rotation, thumb gives the direction of ω). For a particle at position vector r from a point on the axis, moving with linear velocity v, the relation is v = ω × r. Taking magnitudes, v = ωr sinθ; if r is measured perpendicular to the axis (θ = 90°), this reduces to the familiar v = ωr.

Angular acceleration α = dω/dt is likewise a vector along the axis (or opposite to ω if the rotation is slowing down).

Torque (moment of force) measures the turning effect of a force about a point: τ = r × F, where r is the position vector from the point to where the force acts. Its magnitude is τ = rF sinθ = F × moment arm (the perpendicular distance from the point to the line of action of F). Torque is maximum when F ⊥ r and zero when F acts along r.

Angular momentum of a particle about a point is l = r × p, with magnitude l = rp sinθ = mvr sinθ — the rotational counterpart of linear momentum.

Differentiating l = r × p: dl/dt = (v × mv) + (r × F). The first term vanishes (v is parallel to itself), leaving dl/dt = τ — the rotational analogue of Newton's second law. For a system, dL/dt = τ_ext (internal torques cancel in pairs). If τ_ext = 0, L stays constant — the law of conservation of angular momentum, seen in a spinning skater pulling in her arms, or a planet speeding up near the Sun.

A rigid body is in mechanical equilibrium when both its linear momentum and angular momentum stay constant — in practice, when it has neither linear acceleration nor angular acceleration. This needs two independent conditions: (i) translational equilibrium: the vector sum of all external forces is zero, ΣF = 0; (ii) rotational equilibrium: the vector sum of all external torques about any point is zero, Στ = 0.

A body can satisfy one condition without the other. Two equal and opposite forces acting along different lines (not through the same point) give ΣF = 0 but a net torque — this pair is called a couple, and it produces pure rotation with no translation (e.g., turning a steering wheel or tap). Conversely, a body can have zero net torque about its COM but still accelerate translationally if ΣF ≠ 0.

For a rigid body resting under gravity, the centre of gravity is the point where the total weight can be assumed to act; it coincides with the centre of mass when the gravitational field is uniform over the body, which is an excellent approximation for all bodies of ordinary size.

Moment of inertia (I) about a given axis is the rotational analogue of mass — the body's resistance to a change in its rotational motion about that axis. For discrete particles, I = Σm_ir_i², with r_i the perpendicular distance from the axis; for a continuous body, I = ∫r² dm.

Unlike mass, I is not a fixed body property — it depends on the axis and on how mass is distributed relative to it (r² dependence means distant mass counts far more). A ring and disc of equal mass and radius differ: the ring has all its mass at the rim (I = MR²), the disc has mass spread inward (I = ½MR²), so the ring is harder to spin up or slow down.

The radius of gyration k is defined by I = Mk², i.e., k = √(I/M) — the distance at which the whole mass, if concentrated, would give the same I.

Perpendicular axis theorem (plane lamina only): I_z = I_x + I_y, where x, y are two mutually perpendicular in-plane axes and z is perpendicular to the plane, all meeting at one point. Proof: a particle m_i at (x_i, y_i) is at distance r_i² = x_i² + y_i² from the z-axis, so I_z = Σm_ir_i² = Σm_ix_i² + Σm_iy_i² = I_y + I_x.

Parallel axis theorem (any rigid body): I = I_cm + Ma², where a is the perpendicular distance between the given axis and a parallel axis through the COM.

Together, these let you build the moment of inertia of a body about any axis once you know it about one convenient axis — used constantly in numericals, such as the MI of a disc about a tangent.

Rotational kinematics runs exactly parallel to linear kinematics, with θ replacing displacement, ω replacing velocity, and α replacing acceleration. For constant angular acceleration α: ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ. These are derived exactly as the v = u + at family is derived, by integrating dω/dt = α (constant) and dθ/dt = ω.

These equations apply directly to a flywheel, a rotating fan slowing to a stop under a constant retarding torque, or a wheel accelerating uniformly from rest — the same three-equation toolkit used throughout Class 11 mechanics, just relabelled.

For a rigid body rotating about a fixed axis under external torque τ, Newton's second law takes the rotational form τ = Iα — the central working equation of the chapter, and the direct analogue of F = ma.

Work and power: a torque τ rotating a body through dθ does work dW = τ dθ, so W = τθ for constant torque, and P = τω. Rotational KE of a body spinning at ω is KE_rot = ½Iω², found by summing ½m_iv_i² = ½m_iω²r_i² over all particles. The work-energy theorem then reads: work done by the net external torque equals the change in rotational KE.

For a rigid body rotating about a fixed axis, the total angular momentum about that axis is L = Iω — the rotational analogue of p = mv. Combined with dL/dt = τ_ext from section 6.7, and I being constant for a rigid body, this gives I(dω/dt) = τ, i.e., τ = Iα, consistent with section 6.12.

If no external torque acts on a rotating rigid body, L = Iω stays constant, but I and ω can individually change as the body's mass distribution changes (a spinning skater pulling arms in decreases I, so ω increases to keep Iω constant — angular speed increases even though no torque is applied).

Rolling without slipping combines translation of the COM with rotation about an axis through the COM. The contact point with the ground is instantaneously at rest, giving the rolling condition v_cm = Rω.

Kinetic energy of rolling: total KE = ½Mv_cm² + ½I_cmω². Writing I_cm = Mk² and ω = v_cm/R: KE = ½Mv_cm²(1 + k²/R²).

Rolling down an incline of height h (friction here does no work and causes no slipping, so energy is conserved): Mgh = ½Mv²(1 + k²/R²) ⟹ v = √[2gh/(1 + k²/R²)], and a = g sinθ/(1 + k²/R²). Since k²/R² is ½ for a disc, ⅖ for a solid sphere and 1 for a ring, the solid sphere reaches the bottom fastest, then the disc, then the ring — independent of mass or radius, since M and R cancel out.