System of Particles and Rotational Motion notes

This chapter extends mechanics from a single particle to a system of particles and rigid bodies. In translational motion, every point of a body moves equally, while in rotational motion, different points move in circles about an axis. The centre of mass represents the average position of mass and helps replace a complex system by a single point for translational motion. Rotational motion introduces angular displacement, angular velocity, torque, angular momentum and moment of inertia. Newton’s laws get rotational forms such as τ = Iα. The chapter also covers rolling motion and equilibrium of rigid bodies. For NEET, this is a high-value chapter with frequent questions on centre of mass, torque, moment of inertia, rolling and angular momentum conservation.

A system of particles may contain many masses moving differently, but its overall translational motion can be described using a special point called the centre of mass. Centre of mass is the mass-weighted average position of all particles. For two particles, it lies closer to the heavier mass. For multiple particles, coordinates are found using weighted averages. For continuous bodies, summation becomes integration. In symmetric bodies of uniform density, centre of mass lies at the geometric centre. The motion of centre of mass depends only on external forces, not internal forces. This idea simplifies explosions, collisions, recoil and motion of extended bodies in NEET problems.

The linear momentum of a system is the vector sum of momenta of all particles. It is also equal to total mass multiplied by velocity of the centre of mass. Newton’s second law for a system states that the rate of change of total momentum equals the net external force. Internal forces occur in equal and opposite pairs, so they cancel in the total momentum equation and cannot change the total momentum of an isolated system. Therefore, when net external force is zero, linear momentum is conserved. This explains recoil of a gun, explosion of bodies and basic rocket propulsion. In rockets, gases expelled backward give the rocket forward momentum.

Torque, also called moment of force, measures the ability of a force to produce rotation about an axis. Its magnitude is τ = rF sinθ, where r is the position vector from axis to point of application and θ is the angle between r and F. Torque direction is given by the right-hand rule. A couple consists of two equal and opposite parallel forces with different lines of action and produces pure rotation. Angular momentum is rotational momentum, defined as L = r × p for a particle and L = Iω for a rigid body rotating about a fixed axis. Net external torque equals rate of change of angular momentum. If external torque is zero, angular momentum is conserved.

Moment of inertia is the rotational analogue of mass. It measures how difficult it is to change the rotational motion of a body about a given axis. Unlike mass, moment of inertia depends not only on the amount of matter but also on how far the matter is distributed from the axis. Mathematically, I = Σmr². Radius of gyration is the distance at which the whole mass could be imagined to be concentrated to give the same moment of inertia. The parallel axis theorem shifts moment of inertia from a centre of mass axis to a parallel axis. The perpendicular axis theorem applies to plane laminae. NEET frequently asks standard moments of inertia of ring, disc, rod, cylinder and sphere.

Rotational kinematics describes rotation without discussing its cause. Angular displacement measures the angle swept by a radius vector, angular velocity is its rate of change and angular acceleration is the rate of change of angular velocity. For constant angular acceleration, equations similar to linear motion apply: ω = ω0 + αt, θ = ω0t + 1/2αt² and ω² = ω0² + 2αθ. Linear and angular quantities are connected by s = rθ, v = rω and a_t = rα. Period is time for one revolution and frequency is revolutions per second. Rolling motion begins when translation and rotation combine, and pure rolling has v_cm = Rω.

Rotational dynamics studies the causes of rotational motion. Just as net force produces linear acceleration, net torque produces angular acceleration according to τ = Iα for fixed-axis rotation. Work done in rotating a body is related to torque and angular displacement, and power in rotational motion is P = τω. A rotating rigid body has rotational kinetic energy 1/2 Iω². Rolling without slipping combines translational kinetic energy of the centre of mass and rotational kinetic energy about the centre of mass. This explains wheels, pulleys and rolling bodies on inclines. In pulley systems, tension may produce torque if the pulley has moment of inertia, changing the acceleration compared with a massless pulley.

A rigid body is in equilibrium when it has neither translational acceleration nor angular acceleration. Therefore, two conditions must be satisfied: net external force must be zero and net external torque about any point must be zero. Translational equilibrium prevents linear motion, while rotational equilibrium prevents angular motion. Static equilibrium is the special case where the body remains at rest. Torque analysis is essential for ladders, beams, seesaws, doors and balances. Centre of gravity is the point where the weight of a body may be considered to act. Equilibrium can be stable, unstable or neutral depending on how the body responds to small displacement from its position.