Paper 4, Section I, A

Dynamics and Relativity | Part IA, 2017

Consider a system of particles with masses mim_{i} and position vectors xi\mathbf{x}_{i}. Write down the definition of the position of the centre of mass R\mathbf{R} of the system. Let TT be the total kinetic energy of the system. Show that

T=12MR˙R˙+12imiy˙iy˙iT=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} \sum_{i} m_{i} \dot{\mathbf{y}}_{i} \cdot \dot{\mathbf{y}}_{i}

where MM is the total mass and yi\mathbf{y}_{i} is the position vector of particle ii with respect to R\mathbf{R}.

The particles are connected to form a rigid body which rotates with angular speed ω\omega about an axis n\mathbf{n} through R\mathbf{R}, where nn=1\mathbf{n} \cdot \mathbf{n}=1. Show that

T=12MR˙R˙+12Iω2,T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} I \omega^{2},

where I=iIiI=\sum_{i} I_{i} and IiI_{i} is the moment of inertia of particle ii about n\mathbf{n}.

Typos? Please submit corrections to this page on GitHub.