Mechanics - 3B Scientific

UE105060
3B SCIENTIFIC® PHYSICS EXPERIMENT
For the motions ϕ + = ϕ1 + ϕ2
and ϕ − = ϕ1 − ϕ2
(initially
chosen arbitrarily) the equation of motion is as follows:
L ⋅ ϕ&&
L ⋅ ϕ&&
+
−
+ g ⋅ ϕ
+
+
The solutions
ϕ
ϕ
+
−
= 0
( g + 2k) ⋅ ϕ = 0
= a cos
+
= a cos
−
−
( ω+
t) + b+
sin( ω+
t)
( ω t) + b sin( ω t)
−
give rise to angular frequencies
−
−
g
g + 2k
ω + = und ω − =
(4)
L
L
corresponding to the natural frequencies for in phase or out
of phase motion (ϕ +
= 0 for out of phase motion and ϕ –
= 0
for in-phase motion).
The deflection of the pendulums can be calculated from the
sum or the difference of the two motions, leading to the
solutions
1
ϕ1
=
2
1
ϕ2
=
2
( a cos( ω t) + b sin( ω t) + a cos( ω t) + b sin( ω t)
)
+
( a cos( ω t) + b sin( ω t) − a cos( ω t) − b sin( ω t)
)
+
+
+
+
+
+
+
Parameters a +
, a –
, b +
and b –
are arbitrary coefficients that
can be calculated from the initial conditions for the two
pendulums at time t = 0.
The easiest case to interpret is where pendulum 1 is deflected
by an angle ϕ 0 from its rest position and released at
time 0 while pendulum 2 remains in its rest position.
1
ϕ1
= ⋅
2
1
ϕ2
= ⋅
2
( ϕ ⋅ cos( ω t) + ϕ ⋅cos( ω t)
)
0
+
( ϕ ⋅ cos( ω t) − ϕ ⋅ cos( ω t)
)
0
+
0
After rearranging the equations they take the form
ϕ1
= ϕ0
⋅ cos
ϕ = ϕ ⋅ sin
2
with
0
ω−
− ω
ω∆
=
2
ω+
+ ω−
ω =
2
0
( ω∆t) ⋅cos( ωt)
( ω t) ⋅cos( ωt)
+
∆
This corresponds to an oscillation of both pendulums at
identical angular frequency ω, where the amplitudes are
modulated at an angular frequency ω ∆ . This kind of modulation
results in beats. In the situation described, the amplitude
of the beats arrives at a maximum since the overall
amplitude falls to a minimum at zero.
−
−
−
−
−
−
−
−
−
−
(2)
(3)
(5)
(6)
(7)
(8)
LIST OF APPARATUS
2 Pendulum rods with angle sensor U8404270
1 Transformer 12 V, 2 A, e.g. U8475430
1 Helical spring with two eyelets, 3 N/m U15027
2 Table clamps U13260
2 Stainless steel rods, 1000 mm U15004
1 Stainless steel rod, 470 mm U15002
4 Universal clamps U13255
1 3B NETlog U11300
1 3B NETlab for Windows U11310
1 PC with Windows 98/2000/XP, Internet Explorer 6 or later,
USB port
SET-UP
Fig. 2
Set-up for recording and evaluating the oscillation of two
identical pendulums coupled together by a spring
The set-up is illustrated in Fig. 2.
• Clamp two stand rods of 1000 mm length to a bench so
that they are about 15 cm apart.
• Attach a short stand rod between them as a horizontal
cross member to lend the set-up more stability.
• Attach the angle sensors to the top of the vertical rods
using universal clamps.
• Attach bobs to the end of the pendulum rods.
• Suspend the pendulum rods from the angle sensors
(there are grooves in the angle sensors to accommodate
the hinge pins of the pendulum rods)
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