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Chapter 9: Rotational Motion
Rigid body instead of a particle
Rotational motion about a fixed axis
Rolling motion (without slipping)
Rotational Motion
Angular Quantities
Kinematical variables to describe the rotational motion:
Angular position, velocity and acceleration
l

( rad )
R
 d
  lim

t 0 t
 dt
( rad/s )
 ave
 d
  lim

t 0 t
 dt
2
( rad/s )
 ave
Rotational Motion
“R” from the Axis (O)
Solid Disk
Solid Cylinder
Rotational Motion
Linear and Angular Quantities
 l
(1)      t (like l  v t )
 R
 l  ( R )t
 v  R
dv
d
(2) a tan 
R
dt
dt
 a tan  R
(3) a rad
v 2 (R)2


  2R
R
R
Rotational Motion
atan
arad
Kinematical Equations
Conversion : x   , v   , a  
1 2
(1)    0   0t  t
2
(2)    0  t
(3)     2 ( -  0 )
2
2
0
Note :   constant
Rotational Motion
Chapter 10:
Rotational Motion (II)
Rigid body instead of a particle
Rotational motion about a fixed axis
Rotational dynamics
Rolling motion (without slipping)
Rotational Motion
Angular Quantities: Vector
Kinematical variables to describe the rotational motion:
Angular position, velocity and acceleration
z
Vector natures
l

( rad )
R
d ˆ
ˆ
k
k ( rad/s )
dt
d ˆ
ˆ
k
k ( rad/s 2 )
dt
R.-H. Rule
y
x
Rotational Motion
Rotational Dynamics: t
(a)
ax
la
(b)
lb

m
I
a x  F cos
 F cos  m a x
  F ;   l    F l
F  l  I 
 t (torque)  F  l (magnitude )
Rotational Motion
Note: t = F R sin
Lever arm : l or R
(Perpendic ular distance ...)
t  F ( R sin )  F  R
t  (F sin ) R  F  R
Rotational Motion
Note: sign of t
t 1  F1 ( R1 sin 90 )
 (50.0 N)(0.300 m)  15.0 N  m
t 2  F2 ( R2 sin 60 )
 (50.0 N)(0.500 m)(0.866)
 21.7 N  m
t net  t 1 (c.c.w )  t 2 (c.w.)
 t 1 (1)  t 2 (1)
 (15.0 N  m) - (21.7 N  m)
 - 6.7 N  m  6.7 N  m (c.w.)
Rotational Motion
Rotational Dynamics: I
F  m a  m ( R  )  t (torque)  F  R  (m R 2 ) 
(a)  I  m R
2
(b)  I   m i Ri
2
(moment of inertia
(moment of inertia
for single particle)
for a group of particles)
m2
m1
Rotational Motion
m3
Rotational Dynamics: I
I  I cm (moment of intertia about the c.m.)
d
I  I cm  Md 2 (parallel - axis theorem)
Rotational Motion
Parallel-axis Theorem
d
I  I cm  Md
2
1
2
2
 MR 0  MR 0
2
3
2
 MR 0
2
Rotational Motion
Parallel-axis Theorem
2
1
l
1
l
2
2
Ig  If  M    M l  M  M l
4 3
 2  12
2
Rotational Motion
Example 1
Calculate the torque on the 2.00-m long
beam due to a 50.0 N force (top) about
(a) point C (= c.m.)
(b) point P
Calculate the torque on the 2.00-m long
beam due to a 60.0 N force about
(a) point C (= c.m.)
(b) point P
Calculate the torque on the 2.00-m long
beam due to a 50.0 N force (bottom) about
(a) point C (= c.m.)
(b) point P
Rotational Motion
Example 1 (cont’d)
Calculate the net torque on the 2.00-m
long beam about
(a) point C (= c.m.)
(b) point P
Rotational Motion