Paper 4, Section II, B

(a) A rigid body $Q$ is made up of $N$ particles of masses $m_{i}$ at positions $\mathbf{r}_{i}(t)$. Let $\mathbf{R}(t)$ denote the position of its centre of mass. Show that the total kinetic energy of $Q$ may be decomposed into $T_{1}$, the kinetic energy of the centre of mass, plus a term $T_{2}$ representing the kinetic energy about the centre of mass.

Suppose now that $Q$ is rotating with angular velocity $\boldsymbol{\omega}$ about its centre of mass. Define the moment of inertia $I$ of $Q$ (about the axis defined by $\boldsymbol{\omega}$ ) and derive an expression for $T_{2}$ in terms of $I$ and $\omega=|\omega|$.

(b) Consider a uniform rod $A B$ of length $2 l$ and mass $M$. Two such rods $A B$ and $B C$ are freely hinged together at $B$. The end $A$ is attached to a fixed point $O$ on a perfectly smooth horizontal floor and $A B$ is able to rotate freely about $O$. The rods are initially at rest, lying in a vertical plane with $C$ resting on the floor and each rod making angle $\alpha$ with the horizontal. The rods subsequently move under gravity in their vertical plane.

Find an expression for the angular velocity of rod $A B$ when it makes angle $\theta$ with the floor. Determine the speed at which the hinge strikes the floor.

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