Transcript Document

Rotation of Rigid Bodies
Rotational Motion: in close analogy with linear motion
(distance/displacement, velocity, acceleration)
Angular measure in “natural units”
Angles and Rotation in radians
r
q
s
Angle = arc length / radius
s
θ
r
from one complete circuit, 360o = 2p rad
45o = p/4 rad
90o = p/2 rad
180o = p rad
1 rad = 57.30o
Phys211C9 p1
Angular velocity
an object which rotates about a fixed axis has an average
angular velocity wav :
q2  q1 q
wav 

t2  t1 t
usually rad/s but sometime rpm, rps
instantaneous angular velocity is given by:
q dq
w  lim

t 0 t
dt
since s  rq
r
s
q
ds
dq
r
or v  rw
dt
dt
Phys211C9 p2
Angular Acceleration: the rate of change of angular speed
ω
ω dω d 2θ
αav 
;
α  lim

 2
t 0 t
t
dt dt
related to linear accelerati on in circular motion :
dv
dω
atan 
r
 atan  rα
dt
dt
v2
arad   w2 r
r
ac
=w2r
total linear acceleration
a  ac  aT
2
2
aT=ar
Phys211C9 p3
Example: the angular position of a flywheel is given by q = (2.00 rad/s3) t3. The diameter of
this flywheel is .360 m.
Find the angular displacement at 2.00s and at 5.00s.
Find the average angular velocity between 2.00s and 5.00s.
Find the instantaneous angular velocity at 2.00s, 3.50s and 5.00s.
Find the average angular acceleration between 2.00s and 5.00s.
Find the instantaneous angular acceleration at 3.50s.
Calculate the speed of a point on the edge of the flywheel at 3.50s.
Calculate the tangential and radial acceleration of a point on the edge of the flywheel at
3.50s.
Phys211C9 p4
Rotation with constant angular acceleration (just like linear 1-d)
Angular
Linear
1 2
q  q0  w0t  at
2
 w0  w 
q  q0  
t
 2 
1 2
x  x0  v0t  at
2
 v0  v 
x  x0  
t
 2 
w  w0  at
v  v0  at
w2  w0  2a(q  q0 )
v 2  v0  2a ( x  x0 )
2
2
watch units!!!
Phys211C9 p5
Example: A wheel with an initial angular velocity of 4.00 rad/s undergoes a constant
acceleration of -1.20 rad/s2.
What is the angular displacement and angular velocity at t= 3.00s?
How many rotations does the wheel make before coming to rest?
Example: A discus thrower turns with an angular acceleration of 50 rad/s2, moving the discus
around a constant radius of .800 m. Find the tangential and centripetal acceleration when the
discus has an angular velocity of 10 rad/s.
Phys211C9 p6
Example: An airplane propeller is to rotate at maximum of 2400 rpm while the aircraft’s
forward velocity is 75.0 m/s.
How big can the propeller be if the the speed of the tips relative to the air is not to
exceed 270 m/s?
At this speed, what is the acceleration of the propeller tip?
Example: discus chain-linked gears, belt drives etc: linear velocity vs angular velocity.
Phys211C9 p7
v
Rotational Kinetic Energy
for a single point particle
1 2 1 2 2
KE  mv  mr w
2
2
r
m
m1
v1
for a solid rotating object, piece by piece
1
1
2
KE  m1 v 1  m2 v 2 2 
2
2
1
1
2 2
2
 m1 r1 w  m2 r2 w 2 
2
2
1
 ( m1 r1 2  m2 r2 2  )w 2
2
1
1 2
2
2
  mr w  Iw
2
2
v3
r1
m2
r2
r3
m3
v2
I   mr 2 
1 2
KE  Iw
2
Phys211C9 p8
Example: Three masses are connected by light bracing as shown. What
is the moment of inertia about each of the axis shown? What would the
kinetic energy for be for rotation at 4.00 rad/s about each of the axis
shown?
.30m
.10kg
.20kg
.30kg
.40m
axis perpendicular to
plane
Phys211C9 p9
Moments of inertia for some common geometric solids
L
R2
L
R2
R
1
I  ML2
3
T hin Rod (axis at end)
1
ML2
12
T hinRod
I
1
I  MR 2
2
Solid Disk
a
1
2
2
M ( R1  R2 )
2
Hollow Cylinder
I
a
R
b
b
1
I  Ma 2
3
Thin Rectangula r Plate (about edge)
1
I  M (a 2  b 2 )
12
Rectangular P lat e(t hroughcenter)
R
2
MR 2
5
Solid Sphere
I
I  MR2
T hin Walled HollowCylinder
R
2
MR 2
3
T hin Walled HollowSphere
I
Phys211C9 p10
A cord is wrapped around a solid 50 kg cylinder which has a diameter
of 0.120 m, and which rotates (frictionlessly) about an axis through
its center. A 9.00 N force is applied to the end of the cable, causing
the cable to unwind and the drum (initially at rest) to rotate. After the
cable has unwound a distance 2.00m, determine
the work done by the force,
the kinetic energy of the drum,
the rotational velocity of the drum, and
the speed of the unwinding cable.
9.00 N
Phys211C9 p11
Combining Translation and Rotation
KE = KEtranslation + KErotation = ½mv2 + ½Iw2
A connection for rolling without slipping:
s=q r
v=wr
a = a r,
a : angular acceleration
Gravitational Potential Energy for an extended object
use center of mass: U = mgycm
The Great Race: 2 objects rolling (from rest) down the same incline
1
1
mgh  KE  mv 2  Iw2
lost PE = gained KE
2
2
2
same radius, object with
1 2 1 v 1 2 1 I 2
 mv  I    mv  2 v
the smallest I has most v
2
2 r 2
2r
1
I 
=> wins race
  m  v 2
2
r2 
Phys211C9 p12
A mass m is suspended by a string wrapped around a pulley of radius R and
moment of inertia I. The mass and pulley are initially at rest. After the mass has
dropped a height h, determine the relation between the final speed of the mass
and the given parameters (m, I, h). Examine the special case where the pulley is
a uniform disk of mass M.
Phys211C9 p13
The Parallel Axis Theorem
The moment of Inertia about an axis is related in a simple way
to the moment of inertia about a parallel axis which runs
through the center of mass:
Ip = Icm + MR2
p
a
I cm  mi ( xi  yi )
2
b
2
cm
I p  mi (( xi  a ) 2  ( yi  b) 2 )
 mi ( xi  yi )  mi (a 2  b 2 )
2
2
 2ami xi  2bmi yi
 I cm  MR 2  0  0
Phys211C9 p14
Example: An 3.6 kg object is found to have a moment of inertia of .132 kg m2 about an axis
which is found to be .15m from is center of mass. What is the moment of inertia of this
object about a parallel axis which does go through the objects center of mass?
Example: Find the moment of inertia of a thin uniform disk about an axis perpendicular to
its flat surface, running along the edge of the disk.
Phys211C9 p15
Beyond discrete masses: adventures in vector calculus!
I  mi ( xi  y i )
2
2
  ( x 2  y 2 )dm
2
2
(
x

y
)dV

Phys211C9 p16