Classical Mechanics
Lecture 15
Today’s Concepts:
a) Parallel Axis Theorem
b) Torque & Angular Acceleration
Mechanics Lecture 15, Slide 1
Schedule
 One unit per
lecture!
 I will rely on
you watching
and
understanding
pre-lecture
videos!!!!
 Lectures will
only contain
summary,
homework
problems,
clicker
questions,
Example exam
problems….
Midterm 3 Wed Dec 11
Mechanics Lecture 14, Slide 2
Main Points
Mechanics Lecture 15, Slide 3
Main Points
Mechanics Lecture 15, Slide 4
Main Points
Mechanics Lecture 15, Slide 5
Parallel Axis Theorem
Mechanics Lecture 15, Slide 6
Parallel Axis Theorem
Smallest when D = 0
Mechanics Lecture 15, Slide 7
Clicker Question
A.
B.
C.
D.
A solid ball of mass M and radius is connected to a rod of
mass m and length L as shown. What is the moment of
inertia of this system about an axis perpendicular to the
other end of the rod?
2
1 2
2
2
A) I = MR  ML  mL
5
3
2
1
2
2
B) I = MR  mL  ML2
5
3
R
M
2
1 2
2
C) I = MR  mL
5
3
m
L
58%
1
D) I = ML2  mL2
3
25%
17%
0%
axis
Mechanics Lecture 15, Slide 8
CheckPoint
A ball of mass 3M at x = 0 is connected to a ball of mass M at x = L by a
massless rod. Consider the three rotation axes A, B, and C as shown, all
parallel to the y axis.
For which rotation axis is the
moment of inertia of the object
smallest? (It may help you to
figure out where the center of
mass of the object is.)
A
C
B
y
x
3M
M
L
100% got this right !!!
0 L/4 L/2
Mechanics Lecture 15, Slide 9
Right Hand Rule for finding Directions
Why do the angular velocity and
acceleration point perpendicular to
the plane of rotation?
Mechanics Lecture 15, Slide 11
Clicker Question
A.
B.
C.
D.
A ball rolls across the floor, and then starts up a ramp as
shown below. In what direction does the angular velocity
vector point when the ball is rolling up the ramp?
64%
A) Into the page
14%
14%
7%
B) Out of the page
C) Up
D) Down
Mechanics Lecture 15, Slide 12
A.
Enter Question Text
B.
C.
D.
A ball rolls across the floor, and then starts up a ramp as
shown below. In what direction does the angular
acceleration vector point when the ball is rolling up the ramp?
54%
46%
A) Into the page
0%
0%
B) Out of the page
Mechanics Lecture 15, Slide 13
A.
Enter Question Text
B.
C.
D.
A ball rolls across the floor, and then starts up a ramp as
shown below. In what direction does the angular acceleration
vector point when the ball is rolling back down the ramp?
75%
A) into the page
25%
B) out of the page
0%
0%
Mechanics Lecture 15, Slide 14
Torque
t = rF sin(q )
Mechanics Lecture 15, Slide 15
Mechanics Lecture 15, Slide 16
CheckPoint
In Case 1, a force F is pushing perpendicular on an object a
distance L/2 from the rotation axis. In Case 2 the same force is
pushing at an angle of 30 degrees a distance L from the axis.
In which case is the torque due to the force about the rotation
axis biggest?
A) Case 1
B) Case 2
C) Same
axis
axis
L/2
100% got this right
90o
Case 1
F
L
30o
F
Case 2
Mechanics Lecture 15, Slide 17
CheckPoint
In which case is the torque due to the force about
the rotation axis biggest?
axis
A) Case 1
B) Case 2 C) Same
90o
L/2
F
A) Perpendicular force
means more torque.
B) F*L = torque. L is
bigger in Case 2 and the
force is the same.
C) Fsin30 is F/2 and its
radius is L so it is FL/2 which
is the same as the other one
as it is FL/2.
Case 1
axis
L
30o
F
Case 2
Mechanics Lecture 15, Slide 18
Torque and Acceleration
Rotational “2nd law”
Mechanics Lecture 15, Slide 19
Similarity to 1D motion
Mechanics Lecture 15, Slide 20
Summary : Torque and Rotational “2nd law”
Mechanics Lecture 15, Slide 21
Clicker Question
A.
B.
Strings are wrapped around the circumference of two solid
disks and pulled with identical forces. Disk 1 has a bigger
radius, but both have the same moment of inertia.
C.
Which disk has the biggest angular acceleration?
A) Disk 1
w2
w1
43%
36%
21%
B) Disk 2
C) same
F
F
Mechanics Lecture 15, Slide 22
Clicker/Checkpoint
A.
B.
C.
Two hoops can rotate freely about fixed axles through their
centers. The hoops have the same mass, but one has twice
the radius of the other. Forces F1 and F2 are applied as
shown.
How are the magnitudes of the two forces related if the
angular acceleration of the two hoops is the same?
83%
A) F2 = F1
B) F2 = 2F1
17%
F2
0%
F1
C) F2 = 4F1
Case 1
Case 2
Mechanics Lecture 15, Slide 23
CheckPoint
How are the magnitudes of the two forces related if the
angular acceleration of the two hoops is the same?
A) F2 = F1
F2
F1
B) F2 = 2F1
C) F2 = 4F1
M, R
Case 1
M, 2R
Case 2
B) twice the radius means 4 times the moment of
inertia, thus 4 times the torque required. But
twice the radius=twice the torque for same force.
 4t = 2F x 2R
Mechanics Lecture 15, Slide 24
t = RF sin q
q
q = 900
q = 00
q = 90  36 = 54
Mechanics Lecture 15, Slide 25
t = RF sin q
Direction is perpendicular to
both R and F, given by the
right hand rule
tx = 0
ty = 0
t z = t F t F t F
1
2
3
Mechanics Lecture 15, Slide 26
1
MR 2
2
(i)
I DISK =
(ii)
t = I
1 2
(iii) K = Iw
2
Use (i) & (ii)
Use (iii)
Mechanics Lecture 15, Slide 27
Moment of Inertia
I total = I rod  I sphere
1
2
2
I total = mrod L2rod  msphere Rsphere
 msphere ( Lrod  Rsphere ) 2
3
5
1
2
2
I total = msphere (4 Rsphere ) 2  msphere Rsphere
 msphere (5Rsphere ) 2
15
5
16 2
I total = (   25)msphere ( Rsphere ) 2
15 5
7
I total = (26 )msphere ( Rsphere ) 2
15
Mechanics Lecture 15, Slide 28
Moment of Inertia
 Lrod
 2

t = rF sin q = 



 F sin 90

t = I

 =
 Lrod 

F
 2 

t
I
=
(26
7
)msphere ( Rsphere ) 2
15
Mechanics Lecture 15, Slide 29
Moment of Inertia
RCM =
mrod RCM ,rod  msphere RCM , sphere
mrod  msphere
= 4.5Rsphere
I total = I rod  I sphere
5
1
1
 2

2
I total =  mrod L2rod  mrod ( Rsphere ) 2    msphere Rsphere
 msphere ( Rsphere ) 2 
2
2
 12
 5

1
5
1
 1
 2

2
I total =  msphere (4 Rsphere ) 2  msphere ( Rsphere ) 2    msphere Rsphere
 msphere ( Rsphere ) 2 
5
2
2
 60
 5

 16 5 2 1 
I total =     msphere ( Rsphere ) 2
 60 4 5 4 
Mechanics Lecture 15, Slide 30
Moment of Inertia
5
2



t = rF sin q =  Rsphere  F sin 0


t = I

 =

t
I
=
0
 16 5 2 1 
2
    msphere ( Rsphere )
 60 4 5 4 
Mechanics Lecture 15, Slide 31
Moment of Inertia
I total = I rod  I sphere
1
 2

2
I total =  mrod L2rod  mrod (4Rsphere ) 2    msphere Rsphere
 msphere ( Rsphere ) 2 
 12
 5

Mechanics Lecture 15, Slide 32