System of Particles and Rotational Motion

Up to now every body has been a point. This chapter removes that assumption and asks what changes when a body has size, shape and the ability to spin. It begins with the centre of mass — the single point that moves as though all the mass and all the external force acted there, so that an exploding shell's fragments scatter while their centre of mass sails on undisturbed — and with the linear momentum of a system, whose conservation follows directly. It then builds the machinery of rotation: the vector product, angular velocity with the relation $\vec{v} = \vec{\omega} \times \vec{r}$, torque $\vec{\tau} = \vec{r} \times \vec{F}$, and angular momentum $\vec{l} = \vec{r} \times \vec{p}$, with the rotational form of Newton's second law $\frac{d\vec{L}}{dt} = \vec{\tau}_{ext}$ and the conservation law it implies. The equilibrium of a rigid body supplies the two conditions every ladder, bar and lever problem rests on, and the principle of moments turns them into a tool. Moment of inertia $I = \sum m_i r_i^2$ then plays the role mass plays in straight-line motion, and once the perpendicular and parallel axis theorems are in hand every standard shape becomes tractable. The chapter closes with the full kinematics and dynamics of rotation about a fixed axis — $\omega = \omega_0 + \alpha t$, $\tau = I\alpha$, $KE = \frac{1}{2}I\omega^2$, $P = \tau\omega$ — the conservation of angular momentum that makes a skater speed up as she pulls her arms in, and rolling motion, where translation and rotation finally meet in the condition $v_{cm} = R\omega$. Two topics trimmed from the rationalised syllabus, the axis theorems and rolling motion, are restored in full because JEE Main, JEE Advanced and NEET ask about them every single year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.