Vector Cross Product Rotational Dynamics 2 PHYS116 A01

Vector (Cross) Product, Rotational Dynamics 2 PHYS-116 A-01, 10/12/2012, Lecture 22 Momchil Velkovsky

Rotations and vectors in 3 d Rotations do not commute! ¡ Any motion in 3 d that has a stationary point has a stationary axis: i. e. it is a rotation ¡ Infinitesimal rotations commute: they are described by vectors ¡ Angular velocity is a vector ¡

The vector product

A force acts an object at a point located at the position What is the torque that this force applies about the origin? 1. 2. 3. 4. 5. zero

Torque revisited ¡ Last time: torque with respect to an axis: ¡ Today: torque with respect to a point O as a vector: r How this is related to our previous definition? F z

The four forces shown all have the same magnitude: F 1 = F 2 = F 3 = F 4. Which force produces the greatest torque about the point O (marked by the blue dot)? F 1 1. 2. 3. 4. 5. F 1 F 2 F 3 F 4 not enough information given to decide F 3 O F 2 F 4

A rigid body in motion about a moving axis

We have a moving axis of rotation, but: ¡ ¡ ¡ If it moves with constant velocity, use the Galileo’s relativity principle (inertial reference frame) If it moves with acceleration and goes through the CM, we can still write: If it moves with acceleration AND goes through the CM AND stays parallel to itself AND is a symmetry axis or one of the three “principal axes” (if there is no symmetry) then:

Rolling with and without slipping • Rolling with slipping may be calculated. Slipping makes things worse (for driving and calculations).

Consider the speed of a yo-yo toy

Rolling ball down a ramp v. CM CM r P l q Free body diagram y CM a. CM Fg N P fs x z h

Three ways to solve the rolling ball motion

Three ways to solve the rolling ball motion

Three ways to solve the rolling ball motion

Reading for next time ¡ Chapter 11. 7 – 11. 12