Transcript Lecture 20.TorqueRot..

Torque & Rotational
Inertia
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
In which of the cases shown below
A) F1
is the torque provided by the
B) F3
applied force about the rotation
axis biggest? For all cases the
magnitude of the applied force is
the same.
C) F4
D) all of them
E) none of them
Reading Quiz
In which of the cases shown below
A) F1
is the torque provided by the
B) F3
applied force about the rotation
axis biggest? For all cases the
magnitude of the applied force is
the same.
The torque is t = F d sin q, and
so the force that is at 90° to the
lever arm is the one that will have
the largest torque. Clearly, to
close the door, you want to push
perpendicularly!!
C) F4
D) all of them
E) none of them
Last Time
Began angular motion
Angular position, displacement
Angular speed, velocity
Angular acceleration
Similarities between translation and rotation
Today
Torque
Rotational inertia (moment of inertia)
Rotational kinetic energy
What is torque?
We recognize there is a relationship between
tangential force and making something rotate.
t  rF first, simple definition
Only the Tangential Component
of a Force Causes a Torque
q is angle between
r and F
The Moment Arm
moment arm
t  rF sin q
t  r F
t  r F
Sign convention for torque
according to most textbooks:
t > 0 if the torque causes a CCW
acceleration.
t < 0 if the torque causes a CW
acceleration.
Conceptual Quiz:
You are using a wrench to loosen a rusty nut.
Which of the arrangements below is least effective
in loosening the nut? Force is proportional to
length of vector.
A.
B.
C.
D.
E. not possible to
determine
B
A
C
D
Answer: C
The force vectors are all the same.
The arrangement that is the least
effective is the one with the shortest
moment arm. That is C.
Conceptual Quiz:
A mechanic is finding it very difficult to muster
enough torque to twist a stubborn bolt with a
wrench, and she wishes she had a length of pipe
to place over the wrench handle to increase her
leverage. Will torque be increased if the
mechanic pulls just as hard on a length of rope
tied to the wrench handle?
A)
B)
C)
D)
Yes
No
Only in space.
Not enough information given.
Answer: B (no)
The rope placed in this position
neither increases the force or the
moment arm (length of
application of the force causing
the torque).
Angular Quantities
If the angular velocity of a
rotating object changes, it
has a tangential acceleration:
atan
dv
dw
=
= R
= Ra
dt
dt
Even if the angular velocity is constant,
each point on the object has a centripetal
acceleration:
2
(wR)
v
2
aR =
=
=wR
R
R
2
atan = Ra
2
acp = aR = w R
Torque and Angular Acceleration
Torque and angular acceleration
a  F / m Newton's 2nd law
a F
 
(last time at   r )
r mr
multiply by (r / r )
rF t
r F
2
  


where
I

mr
and I is
2
I
 r  mr mr
called the rotational inertia (or moment of inertia)
t  I
Newton's 2nd law for rotation
Linear and angular quantities
Linear
m
a
F
Angular



I

t
Similarities between linear and
angular motion quantities ***
x q
v 
a 
v  v0  at
  0   t
1
1
x  x0  (v0  v )t q  q 0  (0   )t
2
2
1 2
1 2
x  x0  v0t  at
q  q 0  0 t   t
2
2
2
2
2
2
v  v0  2a ( x  x0 )   0  2 (q  q 0 )
Look at system of particles
t  t i  mi ri 
2
i
for a fixed axis
i
if I   mi ri  m r  m r  m r  ...
2
2
1 1
2
2 2
i
then t  I
for a fixed axis
2
3 3
Kinetic Energy of a Rotating Object
massless rod
1 2 1
2
K  mv  m(r )
2
2
1
K   mr 2   2
2
1 2
K  I  is the
2
rotational energy
I is called
rotational inertia
Kinetic Energy of a Rotating
Object of Arbitrary Shape
1
2
K    mi vi 

i 2
Rotational Inertia
Moment of Inertia
Rotational kinetic energy
1
1
2
2 2
K    mi vi     mi ri  
 i 2

i 2
1
2 2
K    mi ri   
2 i

where I   mi ri
1 I 2  K
2
2
i
I appears to be quite useful!!
The Rotational Inertia
(Moment of Inertia) of a Hoop
I  MR
M
2
The Rotational Inertia
(Moment of Inertia) of a Disk
1
2
I  MR
2
This is almost
certainly an example
in textbook.
Use calculus to find
this value.
I=
ò R dm
2
Rotational Dynamics; Torque
and Rotational Inertia
m R
The quantity I 
inertia of an object.
i
2
i
is called the rotational
The distribution of mass matters here—these two
objects have the same mass, but the one on the left
has a greater rotational inertia, as so much of its
mass is far from the axis of rotation.
Rotational Inertia for Uniform, Rigid Objects
of Various Shapes and Total Mass M
Do not memorize!!
Rotational
Inertia for
Uniform, Rigid
Objects
of Various
Shapes and
Total Mass M
Demos:
Rotational inertia rods
Moment of Inertia wheel
1 2
K  I
2
where I   mi ri 2
If a physical object is available, the
rotational inertia (moment of
inertia) can be measured
experimentally.
Otherwise, if the object can be
considered to be a continuous
distribution of mass, the rotational
inertia may be calculated:
I=
ò R dm
2
The parallel-axis theorem
gives the rotational
inertia about any axis
parallel to an axis that
goes through the center of
mass of an object:
I = I CM + Mh
I CM
I
2
Falling Rod. A thin rod of length
stands vertically on a table. The rod
begins to fall, but its lower end does
not slide. (a) Determine the angular
velocity of the rod as a function of
the angle  it makes with the
tabletop. (b) What is the speed of the
tip of the rod just before it strikes the
table?
Conceptual Quiz:
A figure skater spins around with her
arms extended. When she pulls in her
arms, her rotational inertia
A)
increases.
B)
decreases.
C)
stays the same.
Answer: B, decreases
The mass stays the same, but the
radius decreases for the mass in her
arms. The I must decrease.
Conceptual Quiz
Two spheres have the same
radius and equal masses. One is
made of solid aluminum, and the
other is made from a hollow
shell of gold. Which one has the
bigger rotational inertia about an
axis through its center?
A) solid
aluminum
B) hollow gold
C) same
hollow
solid
same mass & radius
Conceptual Quiz
Two spheres have the same
radius and equal masses. One is
made of solid aluminum, and the
other is made from a hollow
shell of gold. Which one has the
bigger rotational inertia about an
axis through its center?
Rotational inertia depends on
mass and distance from axis
squared. It is bigger for the
shell because its mass is
located farther from the center.
A) solid aluminum
B) hollow gold
C) same
hollow
solid
same mass & radius
Conceptual Quiz:
Two wheels with fixed hubs, each having a
mass of 1 kg, start from rest, and forces are
applied as shown. Assume the hubs and
spokes are massless, so that the rotational
inertia is I = mR2. In order to impart identical
angular accelerations, how large must F2 be?
A)
B)
C)
D)
E)
0.25 N
0.5 N
1.0 N
2.0 N
4.0 N
t
Fr
  2
I mr
Answer: D
The hint on the figure should
help. You want Fr/I to be the
same ratio. Fr/mr2 = F/mr, so F/r
must have the same ratio = 2.