Systems of Particles and Rotational Motion
Class 11 Physics · Physics Part I · Chapter 6
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Summary
This chapter extends mechanics from a point particle to extended bodies, especially rigid bodies whose shape does not change. The centre of mass of a system is the mass-weighted average position, and it moves as if all the mass and the total external force were concentrated there, so its motion obeys Newton's laws even when individual particles move complexly. The total linear momentum of a system equals the total mass times the velocity of the centre of mass and is conserved when no external force acts. The vector (cross) product is introduced, leading to the definitions of torque \(\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}\) and angular momentum \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\); the rotational analogue of Newton's second law is \(\boldsymbol{\tau}=\mathrm{d}\mathbf{L}/\mathrm{d}t\). When the net external torque is zero, angular momentum is conserved. A rigid body is in mechanical equilibrium when both net force and net torque vanish. The moment of inertia \(I=\sum m_ir_i^{2}\) is the rotational analogue of mass, depending on the mass distribution and axis; the radius of gyration and the theorems of perpendicular and parallel axes help compute it. Rotational kinematics mirrors linear kinematics with \(\theta,\omega,\alpha\), rotational kinetic energy is \(\tfrac12 I\omega^{2}\), and \(L=I\omega\) for rotation about a fixed axis.
Centre of mass and its motionLinear momentum of a system of particlesVector product of two vectorsTorque and angular momentumEquilibrium of a rigid bodyMoment of inertia and theoremsKinematics and dynamics of rotation about a fixed axis
Key terms
Centre of mass
The mass-weighted mean position of a system of particles, which moves as if all external force acted there.
Torque
The turning effect of a force about an axis, \(\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}\); also called moment of force.
Angular momentum
The rotational analogue of linear momentum, \(\mathbf{L}=\mathbf{r}\times\mathbf{p}\); for fixed-axis rotation \(L=I\omega\).
Moment of inertia
A measure of a body's resistance to angular acceleration, \(I=\sum m_ir_i^{2}\), depending on the mass distribution and axis.
Radius of gyration
The distance \(k\) from the axis at which the whole mass may be supposed concentrated to give the same moment of inertia, \(I=Mk^{2}\).
Rigid body
An idealised body in which the distances between all pairs of particles remain fixed under applied forces.
Extra questions & answers
Try one from this chapter
The centre of mass of a system moves as if: