Transcript 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?
A
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 .
B C D E) all are equally effective
Two Spheres.
Two uniform solid spheres of mass
M r
0 centers are apart. ( 0
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
K
1 2
mv
2 1 2 massless rod
K
2
mr
2 2
K
1 2
I
2 is the rotational energy ) 2
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
K
Rotational kinetic energy
i
1 2
m v i i
2
i
1 2
m r i i
2 2
K
1 2
i m r i i
2 2 where
I
i m r i i
2 1 2
I
2
K
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
v
2
r T
2
T r
(2
r
Velocities in Rolling Motion
Rotational Kinetic Energy
The kinetic energy of a rotating object is given by
K
= е
i
1 2
m v i i
2 By substituting the rotational quantities, we find that the rotational kinetic energy can be written: 1 2 rotational
K
=
I w
2 A object that has both translational and rotational motion also has both translational and rotational kinetic energy:
K
= 1 2 2
MV
cm + 1 2
I
cm
w
2 Copyright © 2009 Pearson Education, Inc.
A Disk Rolling Without Slipping
Rolling without slipping:
v
r
(Rolling with slipping:
v
r
)
K
translation + rotati on
K
= 1 2
mv
2 1 2
I
2
K
1 2
mv
2 1 2
I
r
2 1 2
mv
2 1
I mr
2
K
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
2
v
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
E
K f
U f
1 2
mv
2 1
I mr
2
mgh
1 2
mv
2 1
I mr
2 at rest 1 2
mv
2 1
I mr
2
v
2
gh
1
I mr
2
low I, high v
v
1/
I U
= 0
Conceptual Quiz
Which object reaches the bottom first?
MR
2
MR
2 2 5
MR
2 2 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/mr 2
will have the highest speed.
v
2
gh
1
I mr
2 Let’s do the experiment. Do we need to do quiz again?
Sphere fastest
5
Disk almost
2
Hoop
MR
2
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) For mass:
F y
T
mg
ma
translation For pulley: i
TR
I
But we have a connection between
a
and : rotation
a R
We can use these three equations to solve the equation of motion, for example for
a
.
y
T
mg
ma
TR
I
TR
T
2
Ia m
2
R m
mg a
1
mg
ma
1
g I mR
2
ma I m R
2 Check to make sure this is a reasonable answer. Is the sign correct? Is it correct when
I
0?
Conceptual Quiz A large spool has a cord wrapped around an inner
R r F
drum of radius r. Two larger radius disks are attached to the ends of the drum. When pulled as shown, the spool will A) Rotate CW so that the CM moves to the right.
B) Rotate CCW so that the CM moves to the left.
C) Rotate CW so that the CM moves to the left.
D) Rotate CCW so that the CM moves to the right.
E) 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.
r
Do giant yo-yo demo
F
Rotate
F
Rotate
v v r
No motion
W W
R
FR
W
Work done on reel by force.
Rotational Kinetic Energy The torque does work through an angle θ: as it moves the wheel
W q
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 ?
A) case (a) B) case (b) C) no difference D) it depends on the rotational inertia of the dumbbell 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 MV
CM
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 ?
A) case (a) B) case (b) C) no difference D) it depends on the rotational inertia of the dumbbell 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.
W
= ?
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
.