Environmental Engineering Reference
In-Depth Information
The cars are free to travel along a horizontal, frictionless, linear
air track. The spring has an equilibrium length
l
, and spring constant
k
.
At time
t
=
0 the car at
x
1
has zero velocity and the car at
x
2
has velocity
v
0
.
x
1
x
2
(a) Show that the centre of mass lies half-way between the two masses.
(b) Use Hooke's Law to obtain expressions for the force on each car in terms
of
x
1
and
x
2
.
(c) Use Newton's Second Law to obtain equations of motion for each of
the two cars. Add these to show that a frame of reference that has
its origin at the centre of mass has zero acceleration and is therefore
inertial.
(d) Calculate the velocity of the centre of mass in terms of
v
0
.
(e) Introduce the relative coordinate
u
=
x
1
−
x
2
−
l
and hence show that
m
u
¨
+
2
ku
=
0.
(f) Show that
u
A
sin
(ωt)
is a particular solution to the equation you
derived in (e) and hence determine an expression for
ω
. Show also that
A
=
v
0
/ω
.
(g) Describe the motion of the two cars as seen in the lab frame.
(h) Making use of Eq. (3.28), show that the total mechanical energy of the
system is fixed and equal to
mv
0
/
2.
=−
3.4 A uniform rope of mass per unit length
λ
is coiled on a table. One end
is pulled straight up with constant velocity
v
. Consider the rate of change
of momentum and show that the force exerted on the end of the rope as a
function of height
y
is given by
λv
2
F
a
=
+
gλy.
What is the total work done in lifting the end of the rope to a height
y
?
Find an expression for the instantaneous power needed to lift the rope. Com-
pare this with the rate of change of the total mechanical energy of the rope
and comment on your result.
3.5 Two particles of mass
m
1
=
5
.
0kg and
m
2
=
10
.
0 kg are moving with veloc-
4
.
0
j
)
ms
−
1
. Calculate the
reduced mass for this system and determine the velocity of the centre of mass
and the momentum of each of the particles as measured in the centre-of-mass
frame.
3
.
0
j
)
ms
−
1
ities
v
1
=
(
2
.
0
i
+
and
v
2
=
(
−
1
.
0
i
+
