MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Physics 8.01
W14D1-2 Tilted Gyroscope Solution
A wheel is at one end of an axle of length d . The axle is pivoted at an angle  with
respect to the vertical. The wheel is set into motion so that it executes uniform
precession; that is, the wheel’s center of mass moves with uniform circular motion with
z -component of precession angular velocity  z . The wheel has mass m and moment of
r
inertia Icm about its center of mass. Its spin angular velocity  s has magnitude  s and is
directed as shown in the figure below. Assume that the gyroscope approximation holds,
 z   s . Neglect the mass of the axle. What is the z -component of the precession
angular velocity  z ? Does the gyroscope rotate clockwise or counterclockwise about the
vertical axis (as seen from above)?
Solution:
The gravitational force acts at the center of mass and is directed downward,
r
Fgravity  mg kφ . Let S denote the contact point between the pylon and the axle. The
contact forces between the pylon and the axle are acting at the point S so they do not
contribute to the torque. Only the gravitational force contributes to the torque. Let’s
choose cylindrical coordinates. The torque about S is
r
r
r
φ  mg(k)
φ  mgd sin  φ,
 S  rS , cm  Fgravity  (d sin  rφ d cos  k)
(1.1)
which is into the page in the above figure. Because we are assuming that  z   s , we
only consider contribution from the spinning about the flywheel axle to the spin angular
momentum,
r
 s   s sin  rφ  s cos kφ
(1.2)
The spin angular momentum has a vertical and radial component,
r
Lspin
 Icm s sin  rφ Icm s cos  kφ.
cm
(1.3)
We assume that the spin angular velocity  s is constant. As the wheel precesses, the
time derivative of the spin angular momentum arises from the change in the direction of
the radial component of the spin angular momentum,
d r spin
dφ
r
d φ
Lcm   Icm s sin 
 Icm s sin 
.
dt
dt
dt
(1.4)
where we used the fact that
dφ
r d φ

.
dt dt
(1.5)
The z -component of the angular velocity of the flywheel about the vertical axis is
defined to be
z 
d
.
dt
(1.6)
Therefore the rate of change of the spin angular momentum is then
d r spin
L   I cm s sin   z φ.
dt cm
(1.7)
The torque about the point S induces the spin angular momentum about S to change,
S 
dLspin
cm
.
dt
(1.8)
Now substitute Equation (1.1) for the torque about S , and Equation (1.7) for the rate of
change of the spin angular momentum into Equation (1.8),
mgd sin  φ  Icm s sin  z φ.
(1.9)
Solving Equation Error! Reference source not found. for the z -component of the
precession angular velocity of the gyroscope yields
2/6/2016 - 2 -
z  
d mg
.
I cm  s
(1.10)
The z -component of the precession angular velocity is independent of the angle  .
Because  z  0 , the gyroscope will such that the direction of the precession angular
r
velocity,    kφ , is in the negative z -direction. That means that the gyroscope
z
precesss in the clockwise direction when seen from above.
Both the torque and the time derivative of the spin angular momentum point in the ̂ direction, indicating that the gyroscope will precess counterclockwise when seen from
above in agreement with the calculation that  z  0 .
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