3.12 The Coriolis Force
45
operates perpendicular to the object’s direction of motion. The centripetal force (per
unit mass), which is a true force, is given by:
Centripetal force = −Ω
2
r
(3.31)
where r is the object’s distance from the centre of the turntable, and Ω = 2π/T
is the rotation rate with T being the rotation period. Per definition the rotation rate
Ω is positive for anticlockwise rotation and negative for clockwise rotation. When
releasing the object, it will fl away on a straight path with reference to the f xed
frame of reference.
In the rotating frame of reference, on the other hand, the object remains at the
same location and is therefore not moving at all. Consequently, the centripetal force
must be balanced by another force of the same magnitude but acting in the opposite
direction. This apparent force – the centrifugal force – is directed away from the
centre of rotation. Accordingly, the centrifugal force is given by:
Centrifugal force = +Ω
2
r
(3.32)
When releasing the object, an observer in the rotating frame of reference will see
the object flyin away on a curved path – similar to that shown in Fig. 3.12.
3.12.3 Derivation of the Centripetal Force
The speed of any object attached to the turntable is the distance travelled over a time
span. Paths are circles with a circumference of 2πr , where r is the distance from the
centre of rotation, and the time span to complete this circle is the rotation period.
Accordingly, the speed of motion is given by:
v =
2π
T
r = Ωr.
(3.33)
During rotation, the speed of parcels remains the same, but the direction of
motion and thus the velocity changes (Fig. 3.14). The similar triangles in Fig. 3.14
give the relation δv/v = δL/r . Since δL is given by speed multiplied by time span,
this relation can be rearranged to yield the centripetal force (per unit mass):
dv
dt
= −
v
2
r
,
(3.34)
where the minus sign has been included since this force points toward the centre of
rotation. Equation (3.31) follows, if we finall insert (3.33) into the latter equation.
45
operates perpendicular to the object’s direction of motion. The centripetal force (per
unit mass), which is a true force, is given by:
Centripetal force = −Ω
2
r
(3.31)
where r is the object’s distance from the centre of the turntable, and Ω = 2π/T
is the rotation rate with T being the rotation period. Per definition the rotation rate
Ω is positive for anticlockwise rotation and negative for clockwise rotation. When
releasing the object, it will fl away on a straight path with reference to the f xed
frame of reference.
In the rotating frame of reference, on the other hand, the object remains at the
same location and is therefore not moving at all. Consequently, the centripetal force
must be balanced by another force of the same magnitude but acting in the opposite
direction. This apparent force – the centrifugal force – is directed away from the
centre of rotation. Accordingly, the centrifugal force is given by:
Centrifugal force = +Ω
2
r
(3.32)
When releasing the object, an observer in the rotating frame of reference will see
the object flyin away on a curved path – similar to that shown in Fig. 3.12.
3.12.3 Derivation of the Centripetal Force
The speed of any object attached to the turntable is the distance travelled over a time
span. Paths are circles with a circumference of 2πr , where r is the distance from the
centre of rotation, and the time span to complete this circle is the rotation period.
Accordingly, the speed of motion is given by:
v =
2π
T
r = Ωr.
(3.33)
During rotation, the speed of parcels remains the same, but the direction of
motion and thus the velocity changes (Fig. 3.14). The similar triangles in Fig. 3.14
give the relation δv/v = δL/r . Since δL is given by speed multiplied by time span,
this relation can be rearranged to yield the centripetal force (per unit mass):
dv
dt
= −
v
2
r
,
(3.34)
where the minus sign has been included since this force points toward the centre of
rotation. Equation (3.31) follows, if we finall insert (3.33) into the latter equation.