Environmental Engineering Reference
In-Depth Information
Cable
B
A
Figure 2.14
Spacecraft connected by a cable.
Solution 2.3.2
We construct three free body diagrams for the problem; one for each
spacecraft and one for the cable as shown in Figure 2.15. The Third Law can be
applied to give F
A
=
F
A
and F
B
=
F
B
. The equation of motion for the cable is
F
B
−
F
A
=
M
C
a
C
,
where M
C
is the mass and a
C
is the acceleration of the centre of mass of the cable.
If we take the limit where we can ignore the mass of the cable then F
A
F
B
,i.e.
irrespective of the relative capabilities of the two winches, each spaceship experi-
ences a force of the same magnitude. Ship A therefore experiences an acceleration
of magnitude
=
F
A
M
A
a
A
=
and B experiences an acceleration
F
B
M
B
=
F
A
M
B
a
B
=
with the directions of the acceleration vectors as shown in Figure 2.15.
a
A
a
C
a
B
A
C
B
F
′
A
F
A
F
B
F
′
B
Figure 2.15
Free body diagram for two spacecraft connected by a cable.
Example 2.3.3
Atwood's machine consists of two masses m
1
and m
2
connected by
an inextensible rope of length l which is slung over a frictionless pulley of negligible
mass (Figure 2.16). Determine the acceleration of the masses and the tension in the
rope (you may assume that the tension is constant throughout the rope
4
).
Solution 2.3.3
The masses are linked by a rope of constant length, which couples
their motion such that y
1
+
y
2
is a constant. Differentiation of
this equation
4
After reading Chapter 4 you might like to convince yourself that this is a good approximation if the
mass of the pulley is small compared to
m
1
and
m
2
.
