Rotational Kinematics
a) Rotational Motion (radians/s) b) Translational Motion  (m/s)
Movement of object's center of mass from one (x,y) position to another (x,y) postion
In order to understand the physics of a rolling object, it's helpful to measure it rotational kinematics in terms of radians.
I. Radian
a) When a round object rotates 1 radian, it has travelled one radii
(unit  radian/sec)
b) One full rotation of a sphere = 360 degrees or .......
2p radians
c) One radian is about ....
57 degrees
d) Converting radians to d
Equation:
And d =
Ex) How many radians does the arc s represent? Number of radians q =
2
Ex 1) A wheel with a radius of 2.0 m rolls 5.0 m along the floor. Find the number of radians rotated through.
q = d/r
2.5 radians
Ex 2) How many radians are subtended by a 0.10 m arc of a circle of radius of 0.40 m?
.25 radians Ex 3) How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m?
= .25 radians [360°/2pradians]
14 degrees
II. Angular Velocity: a) defined  How fast something spins
b) w = radians/sec
c) Linear velocity (v) and Angular velocity w
Formula:
v/r = w
Ex 4) A ball is rolling along the ground at 5.0 m/s. If the ball's radius is .50 m, what is its angular velocity?
5 m/s = w(.50m)
w = 10 radians/sec
d) Rotating object's period
 time
... rotation
Since w = [2p rad.]/T
Ex 5) Find the period for a rolling object with a rotational velocity of 10. rad/sec. Find period.
T = 2p/[10 rad/sec]
T = .63 sec
f = 1.6 Hz
III. Angular Acceleration
Review: Relating Linear motion to Rotational motion d = rq q = distance rotated in radians v = rw
w  angular velocity
So … a = ra
Since a = v^{2}/r then .. a = r^{2}w^{2}/r
a = rw^{2}
Ex 6) A pool ball with a radius of 4.0 cm accelerates from rest to 5.0 m/s in .1 seconds. Find its angular acceleration.
a = Dw/Dt
a = 50. m/s^{2 } a = ra a = 1.3 x 10^{3} rad/s^{2}
d = ____ v = _____ a = _____
Ex 7) A ball with a radius of .10 m starts from rest and accelerates down a 10. m incline and attains a velocity of 8.0 m/s. Find it’s ANGULAR acceleration. Find a then a
Vf^{2} = Vi^{2 }+ 2ad a = 3.2 m/s^{2} a = ra a = 32 rad/s^{2}
©Tony Mangiacapre.,
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