Fakultät für Physik und Geowissenschaften
Physikalisches Grundpraktikum
M16e
“Spring constant and shear modulus”
Tasks
1. Calibrate the precision inductive displacement transducer gauge. Determine the sensitivity of the
displacement transducer.
2. Determine the spring constant cst using the calibrated displacement transducer by means of static
measurements.
3. Determine the dynamic spring constant (cdyn) measuring the period of spring oscillations for
different loads of the spring.
4. Calculate the torsion (shear) modulus of the spring material using cst and cdyn.
Literature
Physics, P. A. Tipler, 3rd Edition, Vol. 1, Chaps. 11-2, 12-1, 12-3, 12-4
Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik, 3.0, 3.1, 3.2, 3.3
The Physics of Vibration and Waves, H. J. Pain, 4th Edition, Wiley 1968, 2 (SHM, Simple Harmonic
Oscillator)
Accessories
Displacement transducer with digital indicator, equipment for calibration including micrometer
screw, digital counter, different springs, mass pieces
Keywords for preparation
- Hooke`s law, stress-strain-diagram, shear stress, shear strain
- Elastic modules of homogeneous isotropic solids, Young’s modulus, shear modulus
- Strain and oscillation of a spring, equation of motion, damped oscillation, fundamental frequency
- Principle of an inductive displacement transducer gauge
http://www.rdpelectrosense.com/displacement/lvdt/lvdt-principles.htm
1
Remarks
Note the hints to the experiment in the laboratory.
In task 1 and task 2 record 20 and 10 (various masses mB ) measurement values, respectively, and
plot these graphically. Determine the slopes of the graphs and calculate the sensitivity S of the
inductive transducer (S = dU/dx, unit V/mm) as well as the spring constant cst (B = dmB /dx = cst /g, g =
9.81 m/s2).
In task 3 calculate the arithmetic mean for cdyn from at least five different measurements (loads mB).
Specify the confidence interval for a confidence level of 95%.
Influence of the spring mass on the oscillation frequency
Potential energy
If the spring (spring constant c) is extended or compressed by x through loading with the mass m it
stores the potential energy
x
x
1
Epot = − ∫ Fx ' d x ′ = ∫ cx ′ d x ′ = cx 2 .
2
0
0
Kinetic energy
The oscillation of a spring (spring mass mF, length of the spring L) can be
characterized as follows:
dl
L
The uniform extension/compression of the spring over the total spring
length leads to a displacement δ of the spring segment dl from the
respective rest position l with
l
δ =x .
L
Spring is thought to be decomposed into segments d mF = mF
Thus the kinetic energy of the spring segment dl is
2
1
1 dl ⎛ l ⎞
dmF δ&2 = mF x& 2 ⎜ ⎟ ,
2
2
L ⎝L⎠
and for the kinetic energy Ekin,F of the spring one obtains
L
Ekin,F =
∫
l =0
mF
2
x& 2
l2
L3
dl =
mF 2 l 3
x& 3
2
3L
2
L
0
1
= mF x& 2 .
6
Since the total kinetic energy is the sum of the kinetic energy of the spring Ekin,F and the kinetic
m
energy of the mass Ekin,m ( Ekin,m = x& 2 ), the law of conservation of energy yields for the total energy
2
2
cx
m ⎞
1⎛
Eges = Ekin + Epot = ⎜ m + F ⎟ x& 2 +
= const .
2⎝
3 ⎠
2
The equation of motion (neglecting the damping) is obtained from
dEges
dt
= 0 as
2c x& x
m ⎞
1⎛
c
m + F ⎟ 2 x& &&
x+
= 0 and &&
x+
x =0.
⎜
m
2⎝
3 ⎠
2
m+ F
3
This leads to the natural angular frequency (eigen angular frequency) of the oscillation
ω0 =
c
m
m+ F
3
.
Determination of the shear modulus G
When the spring is extended the potential energy Ep (see above) is stored by the spring.
r: radius of wire
R: winding radius
ϕ: shear angle
On the other, this potential energy can as well be expressed by the potential energy of torsion (shear
angle ϕ) of the wire
πG r4 2
ϕ
Ep =
4l
where G is the shear modulus of the wire and l its length. The wire length l can be related
approximately to the winding radius R and the number of turns n by l = 2πnR.
For small shear angles the spring extension x and the shear angle ϕ are related by ϕ = x/R, see figure,
such that
G r4 2 G r4 2
ϕ =
Ep =
x .
8 nR
8 nR 3
3