Chapter 6 ยท Introduction

Chapter 6 โ€” System of Particles and Rotational Motion

Real bodies have size: they spin, tip over and balance. The centre of mass, torque, angular momentum and moment of inertia extend Newton's laws from particles to rotating rigid bodies.

Toss a hammer to a friend and watch it closely. The head and the handle cartwheel through the air on wild, looping paths โ€” yet one special point inside the hammer glides along a perfectly smooth parabola, exactly as a small stone thrown the same way would. A diver leaving the springboard does the same: arms and legs whirl, but that one point follows the parabola, and by pulling into a tuck she spins faster without anyone pushing her. Push a heavy door near its hinges and it hardly moves; push at the handle with the same force and it swings open easily.

Until now every body in this course has been treated as a particle. This chapter gives bodies their size. First you will find the centre of mass โ€” the point that moves as if all the mass and all the external forces were concentrated there โ€” and see why an exploding firework or a person walking in a boat cannot move it. Then you will meet the turning effect of a force, torque, and the rotational partner of momentum, angular momentum, built with a new way of multiplying vectors, the vector product. These give the conditions for a rigid body to stay in balance โ€” the physics of see-saws, cranes and ladders. Finally the moment of inertia plays the role of mass in rotation: $\tau=I\alpha$ is Newton's second law for a spinning wheel, and conservation of angular momentum explains the diver, the ice skater and the spinning neutron star.

Rigid bodiesโ†’Centre of massโ†’$M\mathbf{a}_\text{cm}=\mathbf{F}_\text{ext}$โ†’Vector productโ†’Torque $\boldsymbol\tau=\mathbf{r}\times\mathbf{F}$โ†’Angular momentumโ†’Equilibriumโ†’Moment of inertiaโ†’$\tau=I\alpha$โ†’Conservation of $L$
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • distinguish pure translation from rotation about a fixed axis, and describe a rigid body's motion as a combination of the two;
  • locate the centre of mass of a set of particles, of symmetric uniform bodies and of composite bodies (including bodies with holes);
  • use $M\mathbf{a}_\text{cm}=\mathbf{F}_\text{ext}$ and $\mathbf{P}=M\mathbf{v}_\text{cm}$ to predict the motion of a system, including explosions and people moving on boats;
  • calculate a vector product $\mathbf{A}\times\mathbf{B}$ and use it for torque $\boldsymbol\tau=\mathbf{r}\times\mathbf{F}$ and angular momentum $\mathbf{l}=\mathbf{r}\times\mathbf{p}$;
  • apply the two conditions of equilibrium of a rigid body to levers, beams and ladders, and find a centre of gravity;
  • use the standard moments of inertia, the radius of gyration, $\tau=I\alpha$, $K=\tfrac12I\omega^2$ and the equations of uniformly accelerated rotation;
  • use $L=I\omega$ and the conservation of angular momentum about a fixed axis;
  • tackle Board, JEE Main, JEE Advanced and NEET-style problems โ€” with the parallel- and perpendicular-axes theorems (JEE Main, JEE Advanced, NEET) and rolling without slipping (JEE Advanced) as exam extensions.
Teaser โ€” The Tumbling Dumbbell

A light rod $1.2\ \text{m}$ long with a ball at each end is thrown at $12\ \text{m/s}$, $55^\circ$ above the horizontal, and set spinning. The faint copies are snapshots every $0.12\ \text{s}$. Make one ball heavier, spin it faster โ€” and look for the one point whose path never changes. Before you press Play, predict: where on the rod is that point?

3.08 rad/s
centre of mass from $m_1$โ€”
time, height of c.m.โ€”
turns madeโ€”
range of c.m.โ€”

Sections in this chapter

6.1

Centre of Mass and Its Motion

Rigid bodies, the centre of mass of particles and of extended bodies, and how the centre of mass moves.

$M\mathbf{a}_\text{cm}=\mathbf{F}_\text{ext}$
Open section โ†’6.2

Torque, Angular Momentum and Equilibrium of a Rigid Body

The vector product, angular velocity, torque, angular momentum, balance of a rigid body and the centre of gravity.

$\boldsymbol\tau=\mathbf{r}\times\mathbf{F}=\dfrac{d\mathbf{L}}{dt}$
Open section โ†’6.3

Moment of Inertia and Rotation about a Fixed Axis

Moment of inertia, rotational kinematics and dynamics, rotational kinetic energy and conservation of angular momentum.

$\tau=I\alpha,\ L=I\omega$
Open section โ†’
โœ… Before you start
From Chapter 4 you need Newton's second and third laws, internal and external forces and the conservation of momentum (Section 4.1, Section 4.2) and the equilibrium of a particle (Section 4.3). From Chapter 5, kinetic energy, work and power (Section 5.1, Section 5.3) and the scalar product. From Chapter 3, vector components (Section 3.2), projectile motion (Section 3.3) and angular speed in circular motion (Section 3.4). The kinematic equations of Section 2.3 reappear with angles in place of distances.
โ† 5.3 Power and Collisions6.1 Centre of Mass and Its Motion โ†’