Rotational Kinematics and Inertia

Circular Motion Angular displacement  =  2 -  1 è How far it has rotated  Units radians 2  = 1 revolution Angular velocity  =  t è How fast it is rotating  Units radians/second 2  = 1 revolution Angular acceleration is the change in angular velocity  divided by the change in time. α =  t è How much is it speeding up or slowing down è Units radians/second 2 27

Period, Frequency l Frequency è Number of revolutions per sec Period =1/frequency T = 1/f = 2  è Time to complete 1 revolution

Circular to Linear (Why use Radians) Displacement  s = r   in radians) Speed |v| =  s/  t = r  /  t = r   Direction of v is tangent to circle  cceleration |a| = rα 29

Angular Acceleration If the speed of a roller coaster car is 15 m/s at the top of a 20 m loop, and 25 m/s at the bottom. What is the cars average angular acceleration if it takes 1.6 seconds to go from the top to the bottom? 41

Comparison to 1-D kinematics AngularLinear And for a point at a distance R from the rotation axis: x = R  v =  R  a =  R

Example: cd player l The CD in your disk player spins at about 20 radians/second. If it accelerates uniformly from rest with angular acceleration of 15 rad/s 2, how many revolutions does the disk make before it is at the proper speed? 48

Example: 48x cd-rom l A 48x cd-rom spins at about 9600 rpm. If it takes 1.5 sec. to get up to speed, what is the angular acceleration? How many revolutions does the disk make before it is at the proper speed? 48

Rotational Inertia, I l Tells how difficult it is get object spinning. Just like mass tells you how difficult it is to get object moving. è F net = m a Linear Motion è τ net = I α Rotational Motion I =  m i r i 2 (units kg m 2 ) l Note! Rotational Inertia depends on what you are spinning about (basically the r i in the equation). 13

Inertia Rods Two batons have equal mass and length. Which will be “easier” to spin A) Mass on ends B) Same C) Mass in center I =  m r 2 Further mass is from axis of rotation, greater moment of inertia (harder to spin) 21

Example: baseball bat

Rotational Inertia Table For objects with finite number of masses, use I =  m r 2. For “continuous” objects, use table below. 33