Environmental Engineering Reference
In-Depth Information
The net result is a type of motion known as nutation (which means 'nodding')
in which the gyroscope precesses slowly about the lab
k
axis, but not at fixed
θ
. Instead, the axle of the gyroscope weaves an oscillatory pattern (with period
2
π/
) about an average angle of tilt while precessing about the vertical direction
(see Figure 10.21). With a simple gyroscope nutation is often heavily damped due
to friction in the pivot and is then observed as a quickly-decaying, fast oscillation
in the angle of the gyroscope.
w
p
A
B
Figure 10.21 An example of a nutation pattern of a gyroscope as observed in the lab frame.
The time for the gyroscope to travel from
A
to
B
is 2
π/
.
PROBLEMS 10
10.1 A child's hoop of mass
m
and radius
r
rolls without slipping in a straight
line with speed
v
. To change the direction of the hoop, the child taps it with
a stick, applying an impulse
p
that is perpendicular to the plane of the
hoop. Where on the hoop should the child apply the blow?
Show that the hoop is deflected by an angle
p
mv
.
θ
≈
You should assume that the plane of the hoop remains vertical throughout.
10.2 A coin may be rolled without slipping in a circular path on a horizontal
surface, as long as it leans slightly towards the centre of the circle. Show
that the angle of tilt
φ
of the coin is given by
3
v
2
2
gR
,
tan
φ
≈
where
v
is the constant speed of the coin,
R
is the radius of the circle and
g
is the gravitational acceleration. What happens if
v
decreases slowly with
time due to rolling friction?
10.3 A spinning ball of diameter 15 cm may be balanced on a fingertip if the
rotational speed is sufficient. Estimate the spin required for: (a) a uniform
solid ball; (b) a uniform hollow thin-walled ball.