Environmental Engineering Reference
In-Depth Information
i.e.
v
+
ω
×
v
=
r
,
(8.18)
where
v
is the velocity of the particle as determined in the rotating frame. To get
the acceleration we can use Eq. (8.16) again to determine the time rate of change
of
v
,i.e.
d
(
v
+
ω
×
rot
+
ω
×
d
v
d
t
r
)
(
v
+
ω
×
=
r
).
(8.19)
d
t
The acceleration in the rotating frame is
a
=
d
v
/
d
t
and using the product rule in
the case of a vector product allows us to write
a
+
v
+
ω
×
a
=
2
ω
×
(
ω
×
r
).
(8.20)
Equations (8.18) and (8.20) are our final expressions relating the velocity and
acceleration in a rotating frame to the same quantities in an inertial frame. Notice
that if the point is at rest in the rotating frame then
v
=
0
and these equations
reduce to the familiar expressions which relate the velocity and acceleration to the
angular velocity and position vector for a particle undergoing circular motion, i.e.
v
=
ω
×
r
(8.21)
and
a
=
ω
×
(
ω
×
r
).
(8.22)
The second of these is none other than the equation for the centripetal acceleration
of a particle undergoing circular motion which we derived in Section 1.3.4. You
should certainly convince yourself that Eq. (8.22) describes a radial acceleration
of magnitude equal to
ω
2
R
,where
R
is the distance from the axis of rotation.
However, the expressions we have just derived are more general and allow also
for the case where the particle is moving in the rotating frame.
Just as we did in the case of linear acceleration, we can substitute for the accel-
eration
a
in Newton's Second Law in order to derive the fictitious force which acts
in the non-inertial frame, i.e.
v
−
F
fict
=−
2
m
ω
×
m
ω
×
(
ω
×
r
).
(8.23)
The second term on the right hand side is the centrifugal force whilst the first term
is something new. It is called the Coriolis force and it acts upon objects which are
moving within a rotating frame of reference.
Example 8.2.1
Consider the rotating turntable illustrated in Figure 8.3. Show that
(a) a particle rolled radially outwards from the centre will be deflected as illustrated
and (b) that a particle rolled radially inwards from a point on the edge of the
turntable will be deflected in the opposite direction.
