Rotation, angular motion & angular momentom Physics 100 Chapt 6

Rotation

d1d1 d2d2 The ants moved different distances: d 1 is less than d 2

Rotation Both ants moved the Same angle:  1 =  2 (=  )    Angle is a simpler quantity than distance for describing rotational motion

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  velocity v change in d elapsed time = angular vel.  change in  elapsed time =

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration a change in v elapsed time = angular accel.  change in  elapsed time = velocity vangular vel. 

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Moment of inertia = mass x (moment-arm) 2 mass m resistance to change in the state of (linear) motion Moment of Inertia I (= mr 2 ) resistance to change in the state of angular motion M x moment arm

Moment of inertial M M x rr I  Mr 2 r = dist from axis of rotation I=small I=large (same M) easy to turn harder to turn

Moment of inertia

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) torque = force x moment-arm Same force; bigger torque Same force; even bigger torque mass mmoment of inertia I

Teeter-Totter F F but Boy’s moment-arm is larger.. His weight produces a larger torque Forces are the same..

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) mass mmoment of inertia I momentum p (=mv) angular mom. L  (=I  ) Angular momentum is conserved: L=const I  = I 

Conservation of angular momentum II II II

High Diver II II II

Conservation of angular momentum II II

Angular momentum is a vector Right -hand rule

Conservation of angular momentum L has no vertical component No torques possible Around vertical axis  vertical component of L= const Girl spins: net vertical component of L still = 0

Turning bicycle L L These compensate

Torque is also a vector wrist by pivot point Fingers in F direction F Thumb in  direction another right -hand rule F pivot point  is out of the screen example:

Spinning wheel F  wheel precesses away from viewer

Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) mass mmoment of inertia I momentum p (=mv) kinetic energy ½ mv 2 angular mom. L  (=I  ) rotational k.e. ½ I   I  V KE tot = ½ mV 2 + ½ I  2

Hoop disk sphere race

I I I

I I I KE = ½ mv 2 + ½ I  2

Hoop disk sphere race Every sphere beats every disk & every disk beats every hoop