Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2013

(a) A rigid body QQ is made up of NN particles of masses mim_{i} at positions ri(t)\mathbf{r}_{i}(t). Let R(t)\mathbf{R}(t) denote the position of its centre of mass. Show that the total kinetic energy of QQ may be decomposed into T1T_{1}, the kinetic energy of the centre of mass, plus a term T2T_{2} representing the kinetic energy about the centre of mass.

Suppose now that QQ is rotating with angular velocity ω\boldsymbol{\omega} about its centre of mass. Define the moment of inertia II of QQ (about the axis defined by ω\boldsymbol{\omega} ) and derive an expression for T2T_{2} in terms of II and ω=ω\omega=|\omega|.

(b) Consider a uniform rod ABA B of length 2l2 l and mass MM. Two such rods ABA B and BCB C are freely hinged together at BB. The end AA is attached to a fixed point OO on a perfectly smooth horizontal floor and ABA B is able to rotate freely about OO. The rods are initially at rest, lying in a vertical plane with CC resting on the floor and each rod making angle α\alpha with the horizontal. The rods subsequently move under gravity in their vertical plane.

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

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