This experiment was carried out to determine a value for the acceleration due to gravity g. The periods of a simple pendulum were timed and recorded for each of the ten variations in the len…Full description
experiment sheet for compound pendulumFull description
Descripción: Conceptos generales del esfuerzo de torsión
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Torsion PuentesDescripción completa
Chapter 1
Mechanics and Acoustics
1.4. THE TORSION PENDULUM. PROPAGATION VELOCITY OF TRANSVERSAL WAVES IN SOLID MEDIA
1.
THEO TH EORE RETI TICA CAL L CON CONSI SIDE DERA RATI TION ON
When the metallic wire of a torsion pendulum finds itself under the action of a torque, the force momentum M is is proportional with the torsion angle : M=-C
(1),
where C is is the elastic constant of torsion. From the kinetic moment theory:
M I 0
(2),
where I 0 represents the system's momentum of inertia. From (1) and (2) one obtains the following differential equation:
C I 0
0 or
2 0
C
where with
0
I 0
0
(3),
it is denoted the pendulum internal frequency.
In the approximation of small oscillations, the oscillation period is given by the following relation: T0
2
2
I 0
0
C
(4)
In figure 1 is presented the sketch of a Weber-Gauss torsion pendulum. For two solid bodies of equal mass
m
placed at distance l symmetrically with respect to the
pendulum axis the momentum of inertia becomes: b ecomes: I
I0
2
2ml
(5)
and the oscillation period: T
I 0
2
2
2ml
C
(6)
The elastic constant of torsion can be determined from the relations (3) and (5) and it is expressed in the following form: 2
C
2
8 ml T
2
2 0
T
(7)
The torsion modulus of the material can be determined with the aid of the equation: D
32 L
4
d
C
(8),
19
Chapter 1
Mechanics and Acoustics
where L is the length and d the the wire diameter. Knowing the torsion modulus and the material density from which the wire is made of one can determine de propagation velocity of the transversal waves in the material: vl
D
(9)
2.
EXPE EX PERI RIME MENT NTAL AL SE SETU TUP P
In the figure 1 the experimental device is presented consisting of:
support S
wire W
horizontal rod HR
solid bodies of equal mass m Fig. 1. The Weber-Gauss torsionSpendulum
W L l
l
m
Fig. 1. Experimental setup
3.
DIRECTIONS
(a) with a ruler measure measure the wire length L and with a micrometer the diameter d ; (b) determ determine ine the period period of small small oscillation oscillation (T 0) for the unloaded pendulum by measuring the time t 0 in which the pendulum carries out a number of oscillations (more than 5) ( T 0 determine
t 0 n0
n0 complete
). Repeat this operation at least three times and
T ;
(c) var vary y the the dista distance nce l and and repeat the previous step.
20
Chapter 1
Mechanics and Acoustics
Determine the elastic constant C from from the relation (7) by using instead of T 0 and T the the T0
average values
and T . The torsion modulus can be determined using the relation
(8). Hence, the propagation velocity of the transversal waves will be determined using the relation (9). The measured as well as the calculated values will be transcribed into the experimental data table. 3
For the calculations m=49.7 g and =7880 =7880 kg/m shall be considered.
4.
EXPE EX PERI RIME MENT NTAL AL DA DATA TA TA TABL BLE E
