In this demonstration Laithwaite uses a gyroscope on a long arm that
can rotate about a vertical axis. This arm is supported on a pivot
point that can be clamped, when unclamped the arm is free to move
up and down.
Laithwaite unclamps the pivot point, the gyroscope
is free to
pivot up and down about this point. When the rotor is spun up
precession is observed. Next Laithwaite clamps the pivot
point and no precession is observed. He then attempts to push the
gyroscope around to the same precession speed as before. However, the
gyroscope topples when pushed.
The experiment was
repeated using a stand similar to that of
Videos showing precession with
pivot point unclamped
pivot point was then
clamped with the arm in a horizontal position
pivot point clamped
Videos showing what happens when
the pivot point is clamped and one attempts to push the gyro round
along with other gyroscopic motion can be observed in an interactive
flash animation via the following link: flash
The general motion
assuming frictionless conditions is explained below.
Pivot point unclamped
the pivot point is
unclamped precession of the gyroscope is observed. It can be seen that
there is also a 'wobbling' of the gyroscope up and down, this motion is
known as nutation.
free body diagrams show the forces acting in this case.
precessing with speed, and so the d'Alembert force,
(where R is the radius of the circle of precession, i.e. the distance
between the pivot point and the center of mass of the rotor) is also
can be seen that in the free body diagram of the base
all the forces are coincident and so moment equilibrium and force
equilibrium are satisfied. However, in the free body diagram
the arm it can be seen that force equilibrium is observed, but moment
equilibrium is not.
arm is able to pivot
freely about the pivot point, therefore, the
pivot point cannot transmit a moment and can be modeled as a pin joint.
From the free body diagram of the arm it can be seen that the offset of
of the rotor and reaction results in an external couple. The magnitude of this
couple is given by the distance between the rotors center of mass and
the pivot point times mg.
moment equilibrium does not exist there is an overall couple, Q
if the point P is stationary - as is the pivot point) and so precession can occur. The
gyroscope is held up because the moment of its weight about the support
is balanced by the change in direction of h.
the precession of the gyroscope is slow
is therefore small and the d'Alembert force may be treated as
negligible and ignored. In this case the free body diagrams
can be seen that there is still a net overall couple acting. This is
again due to the fact that the pivot point cannot transmit a moment.
Therefore, as before due to the fact that Q is non zero precession
the precession is fast, then the d'Alembert force is large. The
following is a free body diagram for when is large.
this case it can be seen that the d'Alembert force is large. This has
resulted in the reaction force at the base being outside the base.
Therefore, in this case the stand would topple as the reaction force
must act within the base.
It can be seen that for
fast precession the base must be sufficiently large to ensure that the
reaction force acts within it and toppling does not occur
Pivot point clamped
the pivot point is
clamped precession does not occur.
following is a diagram of the
forces acting when the pivot point is clamped.
be seen that
force and moment equilibrium are observed. The position of the
reaction force is given by p and q which
are determined by the
formula shown on the diagram.
free body diagram of the
arm and clamp can be drawn as follows
arm is clamped
a moment can be transmitted. This moment balances with the moment
caused by the offset of the weight and reaction forces of the rotor.
It can be seen from the above diagram that the overall moment is
zero, moment equilibrium exists, meaning that Q=0. Since there is no
net overall couple there can be no precession. This is observed in
base is too
small, then the reaction force acts outside the base, and so the
stand topples over. The following diagram shows the forces acting
when this is the case.
one attempts to push round the clamped gyro it falls over
the gyroscope is clamped precession does not occur. When one attempts
to push the arm round to the same precession rate as
before (in the case where the stand is large enough that it does not
topple initially) the stand topples. This is because by pushing the arm
changes the direction of spin axis and so the direction of h,
moment of momentum vector. This means that becomes non zero.
fixed point (pivot point) and so a couple is introduced
by pushing the arm.
This couple acts about the base of the stand causing it to topple. It
can be observed that when the arm is clamped and the gyroscope is not
spinning when one pushes the arm it still falls over. This is because
the toppling is caused by applying a sudden impulse to the