Download Lecture 21.Roational..

Rotational Dynamics
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Two forces produce the
same torque. Does it
follow that they have
the same magnitude?
A) yes
B) no
Reading Quiz
Two forces produce the
same torque. Does it
follow that they have
the same magnitude?
A) yes
B) no
Because torque is the product of force times
distance, two different forces that act at different
distances could still give the same torque.
Last Time
Torque
Rotational inertia (moment of inertia)
Rotational kinetic energy – look at again
Today
Rotational kinetic energy - again
Objects rolling – energy, speed
Rotational free-body diagram
Rotational work
Solving Problems in Rotational Dynamics
1. Draw a diagram.
2. Decide what the system comprises.
3. Draw a free-body diagram for each object
under consideration, including all the forces
acting on it and where they act.
4. Find the axis of rotation; calculate the
torques around it.
Copyright © 2009 Pearson Education, Inc.
Solving Problems in Rotational Dynamics
5. Apply Newton’s second law for rotation. If
the rotational inertia is not provided, you
need to find it before proceeding with this
step.  i  I
i
6. Apply Newton’s second law for translation
and other laws and principles as needed.
7. Solve.
8. Check your answer for units and correct
order of magnitude.
Copyright © 2009 Pearson Education, Inc.
Conceptual Quiz
You are using a
wrench to loosen a
rusty nut. Which
arrangement will be
the most effective in
loosening the nut?
A
B
C
D
E) all are equally effective
Conceptual Quiz
You are using a
wrench to loosen a
rusty nut. Which
arrangement will be
the most effective in
loosening the nut?
Because the forces are all
the same, the only
difference is the lever
arm. The arrangement
with the largest lever arm
(case #B) will provide the
largest torque.
A
B
C
D
E) all are equally effective
Two Spheres. Two uniform solid spheres
of mass M and radius r0 are connected by a
thin (massless) rod of length r0 so that the
centers are 3r0 apart. (a) Determine the
moment of inertia of this system about an
axis perpendicular to the rod at its center.
(b) What would be the percentage error if
the masses of each sphere were assumed to
be concentrated at their centers and a very
simple calculation made?
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
Balanced Pole. A 2.30-m-long pole
is balanced vertically on its tip. It
starts to fall and its lower end does
not slip. What will be the speed of
the upper end of the pole just before
it hits the ground? [Hint: Use
conservation of energy.]
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!!
Rotational and
Translational
Motions of a Wheel
In (a) the wheel
is rotating about
the axle.
In (b) the entire
wheel translates
to the right
Rolling Without Slipping
2 r  2 
v

 r  (2 f )r   r
T
 T 
Velocities in Rolling Motion
Rotational Kinetic Energy
The kinetic energy of a rotating object is given by
1
K= å
mi vi2
2
i
By substituting the rotational quantities, we find that
the rotational kinetic energy can be written:
1 2
rotational K = Iw
2
A object that has both translational and rotational
motion also has both translational and rotational
kinetic energy:
Copyright © 2009 Pearson Education, Inc.
1
1
2
2
K = MVcm + I cm w
2
2
A Disk Rolling Without Slipping
Rolling without slipping: v  r
(Rolling with slipping: v r)
K  translation + rotation
1 2 1 2
K = mv  I 
2
2
2
1 2 1 v
1 2
I 
K  mv  I    mv 1  2   K
2
2 r
2
 mr 
Conceptual Quiz:
A disk rolls without slipping along a
horizontal surface. The center of the disk
has a translational speed v. The uppermost
point on the disk has a translational speed of
A)
B)
C)
D)
0
v
v
2v
need more information
Answer: C
We just discussed this. Look at
figure.
An Object Rolls Down an Incline
at rest
U=0
Look at conservation of energy of
objects rolling down inclined plane.
Let U = 0 at bottom.
E  K i  U i  mgh at top
1 2
I 
1 2
I 
E  K f  U f  mv 1  2   0  mv 1  2 
2
2
 mr 
 mr 
1 2
I 
mgh  mv 1  2 
2
 mr 
2 gh
v
I
1 2
low I, high v
v 1/ I
at rest
U=0
Conceptual Quiz
Which object reaches the bottom first?
MR 2
MR 2
2
2
MR 2
5
A) Sphere
B) Solid disk
C) Hoop
D) Same time
Answer:
Remember to look at the value of the
rotational inertia. The value with the
lowest value of I/mr2 will have the
highest speed.
2 gh
v
I
1 2
mr
Let’s do the experiment. Do we need
to do quiz again?
Sphere
2 MR2
5
fastest
Disk
1 MR2
2
almost
Hoop
MR2
slowest
Answer: A
A Mass Suspended from a Pulley
y
Now we also need to draw a rotational free-body diagram,
because objects can rotate as well as translate. (see figure)
 F  T  mg  ma
For pulley:    TR  I 
For mass:
y
i
But we have a connection between a and  :
We can use these three
equations to solve the
equation of motion, for
example for a.
translation
rotation
a

R
y
T  mg  ma
 TR  I
TR  Ia / R  T   Ia / R
Ia m
 2  mg  ma
R m
I 

 mg  ma 1 
2 
 mR 
g
a
I
1
2
mR
 a/R
2
Check to make sure this is a reasonable answer. Is
the sign correct? Is it correct when I  0?
F
Conceptual Quiz
r
A large spool has a cord
R
wrapped around an inner
drum of radius r. Two
larger radius disks are attached to the ends of the
drum. When pulled as shown, the spool will
A)
B)
C)
D)
E)
Rotate CW so that the CM moves to the right.
Rotate CCW so that the CM moves to the left.
Rotate CW so that the CM moves to the left.
Rotate CCW so that the CM moves to the right.
Does not rotate at all, but the CM slides to the
right.
Answer: A
Look at torque. It is into the screen.
See next slide.
Do giant yo-yo demo
F
Rotate
Rotate
F
r
v
v
r
No motion
W  F x
x  R
W  FR
W   
Work done on
reel by force.
Rotational Kinetic Energy
The torque does work as it moves the wheel
through an angle θ:
W = t Dq
Conceptual Quiz
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater
center-of-mass speed ?
A) case (a)
B) case (b)
C) no difference
D) it depends on the rotational
inertia of the dumbbell
Conceptual Quiz
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater
center-of-mass speed ?
Because the same force acts for the
same time interval in both cases, the
change in momentum must be the
same, thus the CM velocity must be
the same.
J = D p = F D t = MVCM
A) case (a)
B) case (b)
C) no difference
D) it depends on the rotational
inertia of the dumbbell
Conceptual Quiz
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater energy ?
A) case (a)
B) case (b)
C) no difference
D) it depends on the rotational
inertia of the dumbbell
Conceptual Quiz
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater energy ?
If the CM velocities are the same, the
translational kinetic energies must
be the same. Because dumbbell (b)
is also rotating, it has rotational
kinetic energy in addition.
D E = W = FD d = ?
A) case (a)
B) case (b)
C) no difference
D) it depends on the rotational
inertia of the dumbbell
Bicycle Wheelie. When bicycle and motorcycle riders “pop a
wheelie,” a large acceleration causes the bike’s front wheel to
leave the ground. Let M be the total mass of the bike-plus-rider
system; let x and y be the horizontal and vertical distance of this
system’s CM from the rear wheel’s point of contact with the
ground (see figure). (a) Determine the horizontal acceleration a
required to barely lift the bike’s front wheel off of the ground.
(b) To minimize the acceleration necessary
to pop a wheelie, should x be
made as small or as large as
possible? How about y? How
should a rider position his or
her body on the bike in order
to achieve these optimal
values for x and y? (c) If
x = 35 cm and y = 95 cm,
find a.