What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Define the centre of mass of a system of particles.

It is the point whose position is the mass-weighted average of the positions of all particles, \(\mathbf{R}=\sum m_i\mathbf{r}_i/\sum m_i\).

How does the centre of mass move?

It moves as though all the mass of the system and the total external force were concentrated at that point.

State the relation between torque and angular momentum.

The net external torque equals the rate of change of angular momentum, \(\boldsymbol{\tau}=\mathrm{d}\mathbf{L}/\mathrm{d}t\).

State the law of conservation of angular momentum.

When the net external torque on a system is zero, its total angular momentum remains constant.

What are the conditions for mechanical equilibrium of a rigid body?

The vector sum of all external forces and the vector sum of all external torques must both be zero.

Define moment of inertia and radius of gyration.

Moment of inertia is \(I=\sum m_ir_i^{2}\); the radius of gyration \(k\) satisfies \(I=Mk^{2}\).

State the theorem of parallel axes.

The moment of inertia about any axis equals that about a parallel axis through the centre of mass plus \(Md^{2}\), where \(d\) is the distance between the axes.

State the theorem of perpendicular axes.

For a planar body, the moment of inertia about an axis perpendicular to its plane equals the sum of the moments of inertia about two perpendicular axes in the plane meeting at that point.

Write the rotational kinetic energy and angular momentum for fixed-axis rotation.

Rotational KE \(=\tfrac12 I\omega^{2}\) and angular momentum \(L=I\omega\).

Systems of Particles and Rotational Motion

Q: Define torque and give its expression.

Torque is the moment of a force about an axis, \(\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}\), with magnitude \(rF\sin\theta\).

Class 11 Physics · Physics Part I · Chapter 6

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CBSE Class 11 · Physics

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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:

Flashcards & interactive practice