国际物理林匹克竞赛试题27-39(英文版）

Figure 1. The instrumentation used.
SAFETY HAZARD: Be careful with the two vertical blades on the large
stand. The blades are sharp!
Photo: Geir Holm
146
THE PHYSICAL PENDULUM
A physical pendulum is an extended physical object of arbitrary shape that can
rotate about a fixed axis. For a physical pendulum of mass M oscillating about
a horizontal axis a distance, l, from the centre of mass, the period, T, for small
angle oscillations is
T
g
I
M l
l = +
2 π
(1)
Here g is the acceleration of gravity, and I is the moment of inertia of the pen-
dulum about an axis parallel to the rotation axis but through the centre of mass.
Figure 2 shows a schematic drawing of the physical pendulum you will be us-
ing. The pendulum consists of a cylindrical metal rod, actually a long screw,
having length L, average radius R, and at least one nut. The values of various
dimensions and masses are summarised in Table 1. By turning the nut you can
place it at any position along the rod. Figure 2 defines two distances, x and l,
that describe the position of the rotation axis relative to the end of the rod and
the centre of mass, respectively.
Figure 2: Schematic drawing of the pendulum
with definition of important quantities.
147
Rod
Length L (400.0 ± 0.4) mm
Average radius R (4.4 ± 0.1) mm
Mass M ROD (210.2 ± 0.2) ?10 -3 kg
Distance between screw threads (1.5000 ± 0.0008) mm
Nut
Height h (9.50 ± 0.05) mm
Depth of groove d (0.55 ± 0.05) mm
Mass M NUT (4.89 ± 0.03) ?10 -3 kg
Table 1: Dimensions and weights of the pendulum
A reminder from the front page: No points will be given for error estimates ex-
cept in 2c. However, it is expected that the correct number of significant fig-
ures are given.
Section 1 : Period of oscillation versus rotation axis position
(4 marks)
a) Measure the oscillation period, T, as a function of the position x, and present
the results in a table.
b) Plot T as a function of x in a graph. Let 1 mm in the graph correspond to
1 mm in x and 1 ms in T. How many positions give an oscillation period equal
to T = 950 ms, T = 1000 ms and T = 1100 ms, respectively?
c) Determine the x and l value that correspond to the minimum value in T.
Section 2 : Determination of g (5 marks)
For a physical pendulum with a fixed moment of inertia, I, a given period, T,
may in some cases be obtained for two different positions of the rotation axis.
Let the corresponding distances between the rotation axis and the centre of
mass be  l 1 and  l 2 . Then the following equation is valid:

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