Transcript Lecture 24.Gyroscope..

Gyroscopes
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
The position vector from the axis to
where a force is applied to a wheel is
r  2j and the force vector is F  3i. In
what direction is the torque?
A)
B)
C)
D)
E)
i
i
j
k
k
k
Answer: E
t = r ´ F = 2 j ´ 3i = - 6k
j
k
j
i
Last Time
Angular momentum and torque
system of particles
rigid objects
Unbalanced torque
Kepler’s 2nd law
Today
Angular momentum and torque
Gyroscopes
Inertial forces, pseudoforces
Review
Struck Think Rod. A
thin rod of mass M and
length rests on a
frictionless table and/4
is
struck at a point
from
its CM by a clay ball of
mass m moving at speed
v. The ball sticks to the
rod. Determine the
translational and
rotational motion of the
rod after the collision.
r
The Right-Hand
Rule for Torque
  r F
r
Torque and Angular Momentum Vectors
  dL
dt
L  τt
r
ω
The Torque Exerted on a Gyroscope
r
Precession of a Gyroscope
The Spinning Top and Gyroscope
A spinning top will
precess around its
point of contact with a
surface, due to the
torque created by
gravity when its axis of
rotation is not vertical.
d

dt
Mgr
=
L
angular velocity
of precession
The Spinning Top and Gyroscope
The angular velocity of the precession is
given by:
Mgr
Mgr
d




L
I
dt
This is also the
angular velocity of
precession of a toy
gyroscope, as shown
previously.
Uses of Gyroscopes
Navigation has largely been superseded
by GPS, but still useful for stabilizers
Satellites – example is Hubble Space
Telescope
Virtual Reality helmets – used to discern
person’s motion
Racing cars – engine is a gyroscope
Bicycles and motor bikes
Robot orientation
Rotating Frames
of Reference;
Inertial Forces
An inertial frame of
reference is one in which
Newton’s laws hold; a
rotating frame of reference
is noninertial, and objects
viewed from such a frame
may move without a force
acting on them.
Rotating Frames of Reference;
Inertial Forces
There is an apparent outward force on
objects in rotating reference frames;
this is a fictitious force, or a pseudo
force. The centrifugal “force” is of this
type; there is no outward force when
viewed from an inertial reference
frame.
Show two videos
Skateboard aerials
Figure skater
Rolling Tire. A boy rolls a tire along a
straight level street. The tire has mass 8.0
kg, radius 0.32 m and moment of inertia
about its central axis of symmetry of 0.83
kg m2. The boy pushes the tire forward
away from him at a speed of 2.1 m/s and
sees that the tire leans 12° to the right.
(a) How will the resultant torque affect the
subsequent motion of the tire?
(b) Compare the change in angular
momentum caused by this torque in 0.20 s
to the original magnitude of angular
momentum.
Conceptual Quiz
A student sits on a spinning stool with her
arms folded. When she extends her arms
away from her body, her kinetic energy
A)
B)
C)
D)
E)
Increases
Decreases
Stays the same
Depends on her mass
Need further information
Conceptual Quiz
B) Decreases
Angular momentum is conserved. Rotational
inertia is increased.
1 2
1
L2
2
K = Iw =
(I w) =
2
2I
2I
increases
How is this possible? What
happened to the energy? Is
conservation of energy violated?
Conceptual Quiz
A student sits on a spinning stool with her
arms folded. When she extends her arms
away from her body, her kinetic energy
B)
Decreases
Why? Some of the kinetic energy is
converted to potential energy, because her
arms are extended!
Conceptual Quiz
The name of the quantity which is greater
for a long 70 kg barbell than for a short 75
kg barbell that makes it harder for us to
hold and twist it is called rotational
A)
B)
C)
D)
E)
kinetic energy
momentum
inertia
mass
center of mass
Conceptual Quiz
The name of the quantity which is
greater for a long 70 kg barbell than for a
short 75 kg barbell that makes it harder
to twist is called rotational
C) inertia
Conceptual Quiz
R
R/6
A wheel of outer radius R has
T’
an axle of radius R/6. Strings
are wrapped as shown around
the rim of the wheel and around the axle. If
the string around the rim of the wheel has
tension T, what must be the tension T’ in the
string around the axle in order for the wheel to
not move?
A) 6T
C) T/6
B) 6Tsinq
D) (T/6)sinq
Answer: A) 6T
We want torque = 0
R
R/6
t = - T ( R) + T '( R / 6) = 0
T ' = 6T
T’
Dropping Disk. A nonrotating cylindrical
disk of moment of inertia I is dropped
onto an identical disk rotating at angular
speed ω. Assuming no external torques,
what is the final common angular speed of
the two disks?