Environmental Engineering Reference
In-Depth Information
T
sin α
T
α
a
m
g
Figure 2.13
A free body diagram for the bob of a conical pendulum.
Solution 2.3.1
The forces acting on the pendulum bob are its weight m
g
and the
tension in the string
T
. We construct a free-body diagram as shown in Figure 2.13
where the force vectors are resolved into vertical and horizontal components. Since
the bob describes a circular orbit in the horizontal plane, the acceleration is hor-
izontal and points towards the centre of the orbit. There is no acceleration in the
vertical direction so
mg
=
T
cos
α.
Applying the Second Law in the horizontal plane we have
mRω
2
,
T
sin
α
=
ma
=
where R
2
π/τ the angular frequency of the orbit and τ
is the period. Substitution for R gives
=
l
sin
α is the radius, ω
=
mlω
2
,
T
=
which can be used to eliminate the unknown tension in the vertical equation to give
g
l
cos
α
.
ω
=
Often we are interested in the motion of several parts of a system, as the following
example illustrates. We divide the system into discrete parts, each with its own
free-body diagram and in so doing we must be careful to identify which body each
force acts upon.
Example 2.3.2
Spacecraft A and B with masses M
A
and M
B
are adrift in outer
space and connected by a cable (see Figure 2.14). Winches on both craft are used to
wind up the cable to reduce their separation. The winch on A is capable of producing
aforceF
A
on the cable and that of B aforceF
B
. Determine the acceleration of
both A and B in the limit that the mass of the cable is negligible compared to the
masses of the spaceships. Ignore gravitational forces.
