AP  Physics  Study  Guide  Chapter  8  &  9:  Rotational  Velocity    v Angular  quantities    

Ø 𝜃𝑟𝑎𝑑 = !!  

Ø 𝑙 = 2𝜋𝑟  Ø Degrees  to  radians:  32.58 = !!"#$

!"#  

Ø One  radian  =  57.3  Ø Angular  velocity    

§ 𝜔 = ∆!∆!  

§ 𝜔!"#$%"$%"&'(# =!"!"  

Ø Angular  acceleration    § Counterclockwise  is  positive    § ∝= ∆!

∆!  

Ø Rotational  to  translational  § ∆𝜃 = 𝜔!𝑡 +

!!𝑎𝑡!  

§ 𝜔!! = 𝜔!! + 2 ∝ ∆𝜃  § ∆𝜃 = !

!𝜔! − 𝜔!  

§ 𝑣! = 𝜔𝑟  § 𝑎 tan = 𝑟 ∝  

§ 𝑎!"!#$ = ∝ 𝑟 ! + !!!

!

!  

§ 𝜔!𝑟 = 𝑎!  v Dynamics  rotational  motion    

Ø Cause  of  rotation  is  torque    Ø 𝜏 = 𝐹𝑑 sin𝜃  Ø 𝜏 = 𝐼 ∝  

§ 𝐼  Is  the  moment  of  inertia  mass  distribution  about  a  rotating  point    Ø 𝐼 =   𝑚𝑟!  Ø 𝜏 = 0  

v Energy    Ø ∆𝐸 = 0  Ø 𝑚𝑔ℎ! =

!!𝑚𝑣!! +

!!𝐼𝜔!  

Ø 𝐾𝐸 = !!𝐼𝜔!  

Ø 𝑣! = 𝜔𝑟  v Rolling  motion    

Ø 𝜏 = ∆!∆!  

Ø 𝐹 = ∆!∆!  

Ø ∆𝐿 = 𝐼!𝜔! − 𝐼!𝜔!  Ø Assume  there  is  no  slipping  “burn  out”  means  we  have  static  friction    Ø Wheel  rolling  is  a  combination  of  rotational  and  translational  motion    Ø Revolutions  is  linear  distance/circumference    

v Angular  dynamics    

Ø 𝜏 = 𝐼 ∝  § 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑠  𝑐ℎ𝑎𝑛𝑔𝑒  𝑖𝑛  𝜔  

Ø 𝐼 = 𝑚𝑟!  § 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟: 𝐼 = !

!𝑚𝑟!  

§ 𝑟𝑜𝑑  𝑎𝑏𝑜𝑢𝑡  𝑠𝑜𝑚𝑒  𝑒𝑛𝑑: 𝐼 = !!𝑚𝐿!  

§ 𝑠𝑜𝑙𝑖𝑑  𝑠𝑝ℎ𝑒𝑟𝑒: 𝐼 = !!𝑚𝑟!  

§ ℎ𝑜𝑜𝑝  𝑜𝑟  𝑝𝑜𝑖𝑛𝑡  𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒: 𝐼 = 𝑚𝑟!    § Assume  density  is  uniform    

Ø 𝐹 = 𝑚𝑎,𝑎 =∝  𝐹 = 𝑚𝑟 ∝  𝐹 𝑥 = 𝑚𝑟! ∝  𝜏 = 𝐹𝑟  

v Torque    Ø Cause  of  rotation  or  motion    Ø 𝜏 = 𝐹𝑟 sin𝜃  Ø 𝜏 = 𝐹!𝑟  

v Kinetic  energy  with  this  stuff    Ø 𝐾𝐸!"#$#%"&$' =

!!𝐼𝜔!  

Ø 𝑣 = !!!!!!

 v Angular  momentum    

Ø Linear  =  𝑝 = 𝑚𝑣  Ø Angular  =  𝐿 = 𝐼𝜔  Ø 𝜏∆𝑡 = 𝐼𝜔! − 𝐼𝜔!