System of Particles and Rotational Motion

The One Thing

The motion of a system or rotating body can be analysed by separating translational motion of its centre of mass from rotational motion about an axis. Torque, moment of inertia and angular momentum provide the rotational counterparts of force, mass and linear momentum, while conservation laws simplify systems with no external force or torque.

Reactions, Processes and Experiments

Key Terms

• System of Particles: A collection of two or more particles considered together for studying their motion and interactions.
• Centre of Mass: The point at which the entire mass of a system may be considered concentrated for analysing translational motion.
• Centre of Mass of Two Particles: For masses m1 and m2 at positions r1 and r2, the centre-of-mass position is rcm = (m1r1 + m2r2)/(m1 + m2).
• Centre of Mass of a System: For particles with masses mi and position vectors ri, rcm = (Σmi ri)/(Σmi).
• Linear Momentum: The product of mass and velocity; for a system, total momentum is the vector sum of the momenta of all particles.
• Torque: The turning effect of a force about a point or axis, given by τ = r × F, with magnitude τ = rF sinθ.
• Moment Arm: The perpendicular distance from the axis of rotation to the line of action of the force.
• Angular Momentum: The rotational equivalent of linear momentum; for a particle, L = r × p, and for a rigid body rotating about a fixed axis, L = Iω.
• Moment of Inertia: The rotational inertia of a body, defined for particles as I = Σmi ri² and dependent on the mass distribution relative to the axis.
• Radius of Gyration: The distance k from the axis at which the entire mass could be assumed concentrated to produce the same moment of inertia; I = Mk².
• Angular Displacement: The angle through which a body rotates, usually measured in radians.
• Angular Velocity: The rate of change of angular displacement, given by ω = dθ/dt.
• Angular Acceleration: The rate of change of angular velocity, given by α = dω/dt.
• Rigid Body: An ideal body in which the distance between every pair of particles remains constant during motion.
• Rotational Equilibrium: A state in which the net external torque on a body is zero, so its angular acceleration is zero.
• Rolling Motion: Combined translational and rotational motion in which a body rolls without slipping when vcm = Rω.

Easily Confused

• Force and torque: Force governs translational motion, whereas torque is the turning effect of a force and governs rotational motion.
• Mass and moment of inertia: Mass measures resistance to translational acceleration, whereas moment of inertia measures resistance to angular acceleration and depends on mass distribution relative to the axis.
• Linear momentum and angular momentum: Linear momentum is p = mv, whereas angular momentum is L = r × p or L = Iω for a rigid body about a fixed axis.
• Centre of mass and geometrical centre: The centre of mass is determined by mass distribution; it coincides with the geometrical centre only for a uniform symmetrical body.
• Rotational equilibrium and complete equilibrium: Rotational equilibrium requires zero net external torque, whereas complete equilibrium requires both zero net external force and zero net external torque.
• Angular speed and linear velocity in uniform circular motion: Angular speed remains constant, but the direction of linear velocity changes continuously.
• Pure rolling and rotation alone: Pure rolling combines translation and rotation and satisfies vcm = Rω; rotation alone does not require this condition.
• Internal and external forces: Internal forces cannot change the motion of the centre of mass, whereas only external forces can do so.
• Thin ring and disc or solid cylinder: For the same mass and radius, a thin ring has I = MR², whereas a disc or solid cylinder has I = 1/2 MR².
• Rod about its centre and rod about one end: The moments of inertia are I = 1/12 ML² and I = 1/3 ML², respectively.

What Gets Asked

• Calculate the centre of mass of two particles or of a system using rcm = (m1r1 + m2r2)/(m1 + m2) or rcm = (Σmi ri)/(Σmi). Marks are lost by failing to use mass-weighted positions.
• Determine the motion of a system’s centre of mass using vcm = (Σmi vi)/(Σmi), P = Mvcm or acm = Fext/M. The key slip is attributing a change in centre-of-mass motion to internal forces.
• Calculate torque using τ = r × F or τ = rF sinθ, and identify when torque is maximum or zero. Marks are lost by using the distance to the point of application instead of the perpendicular distance to the line of action.
• Apply rotational dynamics through τnet = Iα, rotational kinetic energy Krot = 1/2 Iω², work W = τθ or power P = τω. The common error is substituting mass for moment of inertia.
• Use the parallel-axis theorem, perpendicular-axis theorem and standard moments of inertia. Marks are lost by selecting the wrong axis or confusing the values for a thin ring, disc or solid cylinder, sphere or rod.
• Apply conservation of angular momentum using Li = Lf when the net external torque is zero, including cases in which mass distribution or angular speed changes. The key condition must not be omitted.
• Analyse rolling motion using vcm = Rω, acm = Rα and K = 1/2 Mvcm² + 1/2 Icmω². Marks are lost by including only translational or only rotational kinetic energy.