# 5: Rotational Motion, Torque and Angular Momentum

- Page ID
- 17395

- 5.1: Rotation Basics
- So far, we’ve been looking at motion that is easily described in Cartesian coordinates, often moving along straight lines. Such kind of motion happens a lot, but there is a second common class as well: rotational motion. It won’t come as a surprise that to describe rotational motion, polar coordinates (or their 3D counterparts the cylindrical and spherical coordinates) are much handier than Cartesian ones.

- 5.2: Centripetal Force
- ‘Centripetal’ means ‘center-seeking’ (from Latin ‘centrum’ = center and ‘petere’ = to seek). It is important to remember that this is a net resulting force, not a ‘new’ force like that exerted by gravity or a compressed spring.

- 5.3: Torque
- Anyone who has ever used a lever - that is everyone, presumably - knows how useful they are at augmenting force: you push with a small force at the long end, to produce a large force at the short end, and make the crank turn, stone lift or bottle cap pop off.

- 5.4: Moment of Inertia
- In analog with mass representing the inertia of a body undergoing linear acceleration, we’ll identify this quantity as the inertia of a body undergoing rotational acceleration, which we’ll call the moment of inertia and denote by I.

- 5.5: Kinetic Energy of Rotation
- Naturally, a rotating object has kinetic energy - its parts are moving after all (even if they’re just rotating around a fixed axis). The total kinetic energy of rotation is simply the sum of the kinetic energies of all rotating parts, just like the total translational kinetic energy was the sum of the individual kinetic energies of the constituent particles.

- 5.6: Angular Momentum
- We define the angular momentum \(\boldsymbol{L}\) as the rotational counterpart of momentum.

- 5.7: Conservation of Angular Momentum
- Given that the torque is the rotational analog of the force, and the angular momentum is that of the linear momentum, it will not come as a surprise that Newton’s second law of motion has a rotational counterpart that relates the net torque to the time derivative of the angular momentum.

- 5.8: Rolling and Slipping Motion
- You can easily check that when rotating, the object loses much less kinetic energy to work than when sliding - take a water bottle, either on its bottom (sliding only) or on its side (a little sliding plus rolling), push it with the same initial force, and let go: the rolling bottle gets much further. However, somewhat ironically, the bottle can only roll thanks to friction.

- 5.9: Precession and Nutation
- The action of a torque causes a change in angular momentum, as expressed by Equation 5.7.1. A special case arises when the torque is perpendicular to the angular momentum: in that case the change affects only the direction of the angular momentum vector, not its magnitude.