Lecture 19 More on 2 D rotation Rotational

Lecture 19 More on 2 D rotation • Rotational kinetic energy • Rotation and COM motion • Angular momentum with respective to different reference point • Dynamical equation with arbitrary reference point. 1

Rotational kinetic energy around a fixed center • 2

Turn-table example • 3

Dependence of AM and Torque on the reference point • When defining angular momentum L and torque τ, they depend on the reference point. • In general we consider the reference point as the origin of the coordinate. • Sometimes, however, it is convenient to consider the reference point on the Rigid Body itself, such as its center of mass, or any point on the rigid body. • In this case, the dynamical equations are different. 4

A general motion in 2 D • Consider a two dimensional motion of a disk Or 3 D motion of a cylinder • A general motion can be considered as a motion of center of mass and rotation (COM) around the COM • A general motion can also be considered a translational motion of an arbitrary point on the rigid body and rotation around that point. 5

An example • 6

Kinetic energy of rotation in general case General Movement: the COM velocity vc, and rotation around the COM This, however, is not true for any reference point! 7

An example of translation plus rotation 8

Rotation around any point P • We can in fact consider rotation around any reference point P on the rigid body. • It can be shown that the angular velocity around any point P is the same. (this is a challenge!) • However, the dynamical equation with torque becomes different. • One cannot write down a simple energy expression in this case. 9

2 D rotation with a general reference point P • 10

Choosing q as the instant rotation center 11

Choosing COM as the instant rotation center 12

Pulling the wheel solving the acceleration 13