Torque & Rotational Inertia Lecturer: Professor Stephen T. Thornton
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Transcript of Torque & Rotational Inertia Lecturer: Professor Stephen T. Thornton
Torque & Rotational Inertia
Lecturer: Professor Stephen T. Thornton
Reading QuizReading QuizIn which of the cases shown below
is the torque provided by the
applied force about the rotation
axis biggest? For all cases the
magnitude of the applied force is
the same.
A) F1
B) F3
C) F4
D) all of them
E) none of them
Reading QuizReading Quiz
A) F1
B) F3
C) F4
D) all of them
E) none of them
In which of the cases shown below
is the torque provided by the
applied force about the rotation
axis biggest? For all cases the
magnitude of the applied force is
the same.
The torque is == F d sin F d sin
and so the force that is at 90°90°
to the lever arm is the one that
will have the largest torquelargest torque.
Clearly, to close the door, you
want to push perpendicularlyperpendicularly!!
Last TimeBegan 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.
first, simple definitionrF
Only the Tangential Component of a Force Causes a Torque
is angle between and r F
The Moment Arm
sin
r
F
F
F
r
r
moment arm
Sign convention for torque according to most textbooks:
> 0 if the torque causes a CCW acceleration.
< 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
A
C
B
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) Yes B) No C) Only in space. D) 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:
Even if the angular velocity is constant, each point on the object has a centripetal acceleration:
tana Ra=2
cp Ra a Rw= =
tan
dv da R R
dt dt
wa= = =
( )222
R
Rva R
R R
ww= = =
Torque and Angular Acceleration
Torque and angular acceleration
22
/ Newton's 2nd law
(last time )
multiply by ( / )
where and is
called the rotational inertia (or moment of inertia)
Newton's 2nd law for rotatio
t
a F m
a Fa r
r mrr r
r F rFI mr I
r mr mr I
I
n
Linear and angular quantities
Linear Angular
m I
a F
Similarities between linear and angular motion quantities ***
x
v
a
0 0
0 0 0 0
2 20 0 0 0
2 2 2 20 0 0 0
1 1( ) ( )
2 21 1
2 2
2 ( ) 2 ( )
v v at t
x x v v t t
x x v t at t t
v v a x x
Look at system of particles2
2 2 2 21 1 2 2 3 3
for a fixed axis
if ...
then for a fixed axis
i i ii i
i ii
m r
I m r m r m r m r
I
Kinetic Energy of a Rotating Object
massless rod
2 2
2 2
2
1 1( )
2 21
2
1 is the
2
rotational energy
K mv m r
K mr
K I
I is called rotational inertia
Kinetic Energy of a Rotating Object of Arbitrary Shape
21
2 i ii
K m v
Rotational InertiaMoment of Inertia
Rotational kinetic energy2 2 2
2 2
2
1 1
2 2
1
2
where
212
i i i ii i
i ii
i ii
K m v m r
K m r
I m r
I K
I appears to be quite useful!!
The Rotational Inertia (Moment of Inertia) of a Hoop
M
2I MR
The Rotational Inertia (Moment of Inertia) of a Disk
21
2I MR
This is almost certainly an example in textbook.
Use calculus to find this value.
2I R dm=ò
Rotational Dynamics; Torque and Rotational Inertia
The quantity is called the rotational
inertia of an object.
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.
2i iI m R
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 rodsMoment of Inertia wheel
2
2
1
2
where i i
K I
I m r
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:
2I R dm=ò
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:
2CMI I Mh= +
ICMI
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 A) A) solid
aluminum
B) hollow goldB) hollow gold
C) sameC) same
same mass & radius
solid hollow
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?
Conceptual Quiz
same mass & radius
solid hollow
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) A) solid aluminum
B) hollow goldB) hollow gold
C) sameC) same
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) 0.25 N B) 0.5 N C) 1.0 N D) 2.0 N E) 4.0 N
2
Fr
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.