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•
After reaching this angle, reverse the direction
of the string and the direction of the force
again.
•
Increase force, step by step, up to a torsional
angle of 180°.
Once this is done, a complete hysteresis curve
should have been obtained. The characteristic
values of this curve are the torsional angle in the
relieved state (meaning with zero force acting, i.e.
where the curve crosses the x-axis), and the extra
force required to pull the wire back to the original
value of 0° (where the curve crosses the y-axis).
5.3 Dynamic measurement (torsional pendulum)
•
Set up torsion apparatus as in 5.1, but without
the spring scale.
•
Move the circular scale from the point of origin
to an angle of approximately 25° and release
it.
•
Measure the time taken for 10 full, unimpeded
torsional oscillations and take this as a base to
calculate the oscillation period of the system.
•
Conduct an initial measurement with the circu-
lar scale without adding any weight. In subse-
quent measurements, add the two weights
with the 4 mm stems symmetrical either side
the axis of the circular scale and repeat the
measurement. Start at the innermost stem po-
sition, and move the extra weight one position
outward for each measurement.
•
Write down the resulting times, and the dis-
tance of the weights from the rotating axis.
The oscillation period T of the torsional pendulum
can be calculated as:
J
= 2
⋅ π
T
D
where J is the moment of inertia of the oscillator,
and
is the torsion coefficient. The moment of
D
inertia J
tot
the circular scale and the moment of inertia J
the added weights.
J
J
J
=
+
tot
K
If the additional weights are considered, their mo-
ment of inertia can be calculated as:
2
J
= 2mr
C
By squaring equations (1), (2) and (3), this results in:
=
π
2
2
T
4
At first glance, the moment of inertia of the circular
scale remains unknown. According the equation
(4), the plot T
which, due to the circular scale's moment of iner-
tia, does not intersect with the origin.
Rearranging equation (1) for D and substituting (2)
results in:
(
=
+
D
J
K
J
for the circular scale is always constant, J
K
changes, in accordance with (3), corresponding to
the distance of the added weights from the rota-
tion axis. By taking two measurements for the
oscillation period for different J
nated from (5), and D can be calculated with the
following equation:
(
−
J
=
Z
2
D
T
Because of the subtraction, there needs to be a
(1)
large difference between J
error small. An error analysis provides much more
accurate values for dynamic measurements than
for static measurements. In principle, dynamic
measurement results in smaller values for D , since
the friction in static measurements has the effect of
making the torsion coefficient D appear bigger.
3
is the sum of the moment of inertia J
Z
(
)
+
2
π
2
J
2
mr
4
=
+
K
J
K
D
D
2
2
=f(r
) must result in a straight line,
2
π
2
)
J
Z
T
z
)
⋅
π
2
J
4
Z
1
−
2
2
T
2
1
and J
z1
of
K
of
Z
(2)
(3)
π
2
8
m
2
r
(4)
D
(5)
z
, J
can be elimi-
K
(6)
, to keep the
Z2
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