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Physics 211
Lecture 15
Today’s Concepts:
a) Parallel Axis Theorem
b) Torque & Angular Acceleration
Mechanics Lecture 15, Slide 1
Your Comments
every thing was new to me, but not difficult to understand
How many times can Superman run into a wall before he gets a headache?
As long as I went through each step, I arrived at a sensible answer. This is all
easy to do because you just have to walk through the steps. The one thing I
am confused about is direction. So Torque will point along the axis in either
direction. I am just confused on how I can see the apply in real life. Like does
a force pointing along the axis of rotation do anything?
I bet you're going to say this can all be figured out with F = ma huh?
My wish for "To Make a Wish Foundation" was to get this put onto today's
lecture... and it came true...
I discovered that approaching the prelectures with the same zeal as the
homework makes me understand things a lot better. Now, onto serious
business. You don't have Maiden, which is outrageous, but what about Rush?
There is no excuse for not having Rush. Actually, it is a sin. I am expecting
Subdivisions on Thursday.
Can i get a SHOUT OUT?! so i relaized something tonight...F=MA is that awkward
friend that changes his personality depending on who he wants to impress. His
been in 1/2mv^2 and now his in TORQUE?!!!
liking this, seems cpmlicated but once you think about it its okay. Gotta love
the prelectures and garrryyy! cheers, me
Mechanics Lecture 15, Slide 2
Proven Fact:
If you don’t prepare for lecture you don’t learn as much…
You're not supposed to understand everything in the prelecture
– that’s the whole point of having a lecture.
Mechanics Lecture 15, Slide 3
Parallel Axis Theorem
Smallest when D = 0
Mechanics Lecture 15, Slide 4
Clicker Question
A solid ball of mass M and radius is connected to a thin 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
2
1 2
2
C) I = MR  mL
5
3
1
D) I = ML2  mL2
3
R
M
m
L
axis
Mechanics Lecture 15, Slide 5
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
Rotation is smallest around the objects
center of mass, and the center of mass for
this system is at L/4.
0 L/4 L/2
Mechanics Lecture 15, Slide 6
Right Hand Rule for finding Directions
I learned that the right hand rule was
for B fields about wires and B fields
with E fields crossing them. Now you
tell me there is another right hand
rule. Seriously physics, be more
creative. I think that with all the smart
people who study you could at least give
your rules more diverse names.
Mechanics Lecture 15, Slide 7
Clicker Question
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?
A) Into the page
B) Out of the page
C) Up
D) Down
Mechanics Lecture 15, Slide 8
Clicker Question
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?
A) Into the page
B) Out of the page
Mechanics Lecture 15, Slide 9
Clicker Question
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?
A) into the page
B) out of the page
Mechanics Lecture 15, Slide 10
Torque
I quite don't understand
the concept of torque. So
generally, if Torque is
greater, what will happen
other than the door will
closing faster?
t = rF sin(q )
Mechanics Lecture 15, Slide 11
Torque and F=ma
Why in the new equation for torque analogous to Newtons's
Second Law deals with angular acceleration only? Are these
concepts separate?
r
r
F = ma
Fq = maq
Fq = mr
rFq r= mr 
r
r
 r  F = mr 
2
r
r
t
=
I


v
F
q
r
2
Mechanics Lecture 15, Slide 12
Look at Prelecture slide 9 again
It’s all there….
Mechanics Lecture 15, Slide 13
I know torque = rxF but what exactly is r?
r
F
q
r
r
r r
r  F = rF sinq
q
r
F
r
F
q
q
q
r
r
r
r
=r(Fsinq)
=F(rsinq)
Mechanics Lecture 15, Slide 14
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
90o
Case 1
F
L
30o
F
Case 2
Mechanics Lecture 15, Slide 15
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) Pushing perpendicular to
the door is always best..
B) the length of the
rotation arm is longer
in case 2..
C) FL/2=FLsin30
Case 1
axis
L
30o
F
Case 2
Mechanics Lecture 15, Slide 16
Similarity to 1D motion
Mechanics Lecture 15, Slide 17
Mechanics Lecture 15, Slide 18
Clicker Question
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.
Which disk has the biggest angular acceleration?
A) Disk 1
w2
w1
B) Disk 2
C) same
F
F
Mechanics Lecture 15, Slide 19
CheckPoint
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?
A) F2 = F1
B) F2 = 2F1
F2
F1
C) F2 = 4F1
Case 1
Case 2
Mechanics Lecture 15, Slide 20
CheckPoint
How are the magnitudes of the two forces related if the
angular acceleration of the two hoops is the same?
F2
A) F2 = F1
F1
B) F2 = 2F1
M, R
C) F2 = 4F1
Case 1
M, 2R
Case 2
t = I
RF = MR 
F = MR
2
Mechanics Lecture 15, Slide 21
t = R F s in q
q
q = 90 0
q = 00
q = 90  36 = 54
Mechanics Lecture 15, Slide 22
t = R F s in q
Direction is perpendicular to
both R and F, given by the
right hand rule
tx = 0
ty = 0
t z = t F1  t F2  t F3
Mechanics Lecture 15, Slide 23
(i)
1
I DISK = MR2
2
(ii)
t = I
1 2
(iii) K = Iw
2
Use (i) & (ii)
Use (iii)
Mechanics Lecture 15, Slide 24