3-Dimensional Rotation:
Gyroscopes
8.01
W13D2
Torque and Time Derivative of
Angular Momentum
Torque about S is equal to the time
derivative of the angular momentum about S
S
ext
dL S

dt
If the magnitude of the angular momentum
is constant then the torque can cause the
direction of the angular momentum to
change
Time Derivative of a Vector
Consider a vector
where
Ar  A sin 
A  Az kˆ  Ar rˆ
Az  A cos 
A vector can change both
magnitude and direction.
Example: Suppose A  Az kˆ  Ar rˆ
does not change magnitude but
only changes direction then
dA
d ˆ
d ˆ
 Ar
  A sin 

dt
dt
dt
Time Derivative of Vectors
of Constant Length: Circular
Motion
Circular Motion: position vector points radially
outward, with constant magnitude but changes in
direction. The velocity vector points in a tangential
direction to the circle with a constant magnitude. The
acceleration vector points radially inward.
r  r rˆ
dr
d ˆ
v
r

dt
dt
v  r
d
dt
dv
d
 d  ˆ
ˆ
a
 v
r  r 
 r
dt
dt
 dt 
2
Introduction To Gyroscopic
Motion
Deflection of a Free Particle
by a Small Impulse
If the impulse I << p1
the primary effect is to
rotate p about
the x axis by a small
angle  .
Deflection of a Free Particle
by a Small Impulse
I  p  Fave t


L   ave t  r  Fave t
L  r  Favet
L  r  I
The application of I
causes a change in the
angular momentum L
through the torque
equation.
Deflection of a Free Particle
by a Small Impulse
As a result, L rotates about
the x axis by a small angle
. Note that although I is
in the z direction, L is in
the negative y direction.
Effect of a Small Impulse on
a Tethered Ball
The ball is attached to a string rotating about a fixed
point. Neglect gravity.
Effect of a Small Impulse on
a Tethered Ball
The ball is given an impulse perpendicular to r and to p .
Effect of a Small Impulse on
a Tethered Ball
As a result, L rotates about the x axis by
a small angle . Note that although I is in the
z direction, L is in the negative y direction.
Effect of a Small Impulse on
a Tethered Ball
The plane in which the ball moves also rotates about
the x axis by the same angle. Note that although I is
in the z direction, the plane rotates about the x axis.
Concept Question: Effect of
a Large Impulse on a
Tethered Ball
What impulse I must be given to the ball in order
to rotate its orbit by 90 degrees as shown without
changing its speed?
Effect of a Large Impulse on
a Tethered Ball
What impulse I must be given to the ball in order
to rotate its orbit by 90 degrees as shown without
changing its speed?
Solution: Effect of a Large
Impulse on a Tethered Ball
I must halt the y motion and provide a momentum
of equal magnitude along the z direction.
Solution: Effect of a Large
Impulse on a Tethered Ball
L cancels the z component of L and adds a component
of the same magnitude in the negative y direction.
Effect of a Small Impulse
Couple on a Baton
Now we have two equal masses at the ends of a
massless rod which spins about its center. We apply
an impulse couple to insure no motion of the CM.
Effect of a Small Impulse
Couple on a Baton
Again note that the impulse couple is applied in the z
direction. The resulting torque lies along the negative y
direction and the plane of rotation tilts about the x axis.
Effect of a Small Impulse
Couple on Massless Shaft
of a Baton
Instead of applying the impulse couple to the masses one
could apply it to the shaft to achieve the same result.
Concept Question: Effect of
a Small Impulse Couple on
Massless Shaft of a Baton
To make the top of the shaft move in the -y direction
in which direction should one apply the top half of an
impulse couple?
Solution: Effect of a Small
Impulse Couple on
Massless Shaft of a Baton
The impulse couple Ib applied to the shaft has the
same effect as the Ia couple applied directly to the
masses. Both produce a torque in the - y direction.
Effect of a Small Impulse
Couple on Massless Shaft
of a Baton
Trying to twist the shaft around the y axis causes
the shaft and the plane in which the baton moves
to rotate about the x axis.
Effect of a Small Impulse
Couple on a Disk
The plane of a rotating disk and its shaft behave just
like the plane of the rotating baton and its shaft when
one attempts to twist the shaft about the y axis.
Effect of a Small Impulse
Couple on a Non-Rotating
Disc
This unexpected result is due to the large pre-existing L .
If the disk is not rotating to begin with, L is also the
final L . The shaft moves in the direction it is pushed.
Effect of a Small Impulse
Couple on a Disk
It does not matter where along the shaft the impulse
couple is applied, as long as it creates the same torque.
Effect of a Force Couple on
a Rotating Disk
A series of small impulse couples, or equivalently a
continuous force couple, causes the tip of the shaft
to execute circular motion about the x axis.
Effect of a Force Couple on
a Rotating Disk
dL  L  dt

dL
dt
L 
  I


I
The precession rate of the shaft is the ratio of the
magnitude of the torque to the angular momentum.
Precessing Gyroscope
Toy Gyroscope: Forces and
Torque
Gravitational force acts at the center
of the mass and points downward
Contact force between the end of
the axle and the pylon
Torque about the contact point due
to gravitational force
 S  rS ,cm  Fgravity  b rˆ  mg (kˆ )  b mg ˆ
The direction of the torque about
pivot points into the page in the
figure
Torque: Magnitude of Angular
Momentum Changes
If the flywheel of the gyroscope
is not spinning, gyroscope
starts to fall downward and the
torque about the pivot point S
induces the gyroscope to start
rotating about an axis pointing
into page.
d LS
d 2
S 
bmg  I S 2
dt
dt
Torque induces the magnitude
of the angular momentum to
change.
Direction of Angular
Momentum Changes
If the flywheel is spinning, the
spin angular momentum about
the center of mass of the
flywheel points along the axle,
radially outward; the torque
causes the spin angular
momentum to change its
direction, with precessional
angular frequency
d spin
ˆ
ˆ
Lcm  Lspin
cm    I cmS  
dt

d
dt
ˆ
Lspin
cm  Icm S r
d spin
spin d ˆ
Lcm  Lcm

dt
dt
  d / dt
Gyroscope: Precession
Torque about the pivot point
induces the angular
momentum to change
S 
dL S
dt
b mg  Lspin
cm 
Precessional angular
frequency of the gyroscope

b mg
Lspin
cm
b mg

I cm S
Newton’s Second Law: center
of mass remains at rest
Fvertical  mg  0
Frad  mb2
Gyroscopic Approximation
Flywheel is spinning with an
angular velocity
 spin   S rˆ
Precessional angular velocity
   kˆ
Total angular velocity
total    spin
Gyroscopic approximation: the
angular velocity of precession 
is much less than the component
of the spin angular velocity  S ,
 = S
Table Problem: Gyroscope
A gyroscope wheel is at one end of
an axle of length l . The axle is
pivoted at an angle  with respect to
the horizontal. The wheel is set into
motion so that it executes uniform
precession. The wheel has mass m
and moment of inertia Icm about its
center of mass . Its spin angular
velocity is s . Neglect the mass of
the shaft. What is the precessional
frequency of the gyroscope? Which
direction does the gyroscope rotate?