metal wire as measuring element for measuring forces and
temperatures. A treatise on the change of the electrical resistance
of a wire by stretching, bending and warming
by Gerhard Höhne
the resistance of a wire by stretching, by bending and warming change
is a question in which students are very interested, especially then,
if you say, that such changes can be possibly used for electric
measuring of forces and temperatures.
To proof the changes in
resistance, the Wheatstone Bridge is introduced, which is very
important in the technology of measuring. In this case the students
regard it as an impressive invention.
It can be shown easily
using this bridge, that the electrical resistance of a wire increases
by warming and stretching and decreases by bending, and that a wire
of constantan is suitable to measure forces in the range of
10-3 N .
The students are very
surprised about the reduction of the resistance by
bending because they expect, that this should lead to an
increase of the resistance.
Interested students seek an
explanation for the changes in resistance by stretching and bending,
sometimes they wish even a mathematical description. Under these
conditions important topics can be discussed, which are boring under
other circumstances. Combinations of resistances parallel and in
series can be investigated. The resistance of a wire in dependence of
its length and cross-section can be examined and the specific
resistance can be introduced.
1. Examining the change of resistance
by warming, stretching and bending
of the resistance by stretching and bending can easily be
examined using the equipment you can see in fig. 1.
On a circuit
board there is a Wheatstone bridge formed from four 15
Ohm-resistors A, B, C and D. D, the measuringelement is a
constantan-wire (length:90 cm , diameter: 0,2 mm), which hangs freely
between two insulated contacts. The
bridge is attached to a 6V voltage source. The
circuit board still carries an amplifier for the measuring-signal .
If a load is applied to D (look
at fig. 1) , then the attached measuring device shows electrons
flowing from f to e. An explanation for this is immediately found.
The resistors A, B, C and D which had identical resistances
initially are no longer flown through by equal currents,
because due to its extension D has a higher resistance than the other
resistors since it becomes longer and thinner. At first the electrons
coming from the –pole form two equal currents to B and C,
a part of the electrons flowing through C then prefers A to D
and flows through the measuring device, because A has a lower
resistance than D.
The students are very surprised
if they see that a decrease of resistance is indicated to
bending (without simultaneous tension) of wire D. In this case
the current doesn't flow from f to e but from e to f. This mostly
unexpected effect can be made plausible looking at fig. 2.
A piece of a
bent wire is shown in fig. 2, the right and left half can be
treated like two wires switched parallelly. At the bend the
resistance of the left half will be increased by DR
and the one of the right half will be decreased by DR.
It is valid: Two
resistances with a destined sum switched parallelly have the
greatest complete resistance Rg , if the single resistances are
instance the complete resistance of two 15ohm resistances switched
parallelly is greater than the resistance of a combination of a
10ohm- and a 20ohm- wire .
There is no
change in resistance if for instance wire D of constantan is
warmed about 10o
C . This
changes, if instead of a wire of constantan a wire of copper or iron
is being used. The radiant heat of the hand then already causes
a strong deflection of the measuring device. It gets clear, that the
resistance of iron and copper increases at warming.
combination of a wire of manganin and a wire of constantan instead of
D (look at fig. 3), then the thermoelectrical effect can be
demonstrated, too. Touching a
soldered contact manganin-constantan by hand causes a strong
deflection of the measuring device. The deflection then decreases
with the wire measuring transducer
The Wheatstone bridge with a
wire D of constantan can be used as a measuring transducer for
electrical measuring of forces and displacements.
experiment: Addition of forces
demonstration of a wire of constantan as a measuring-element for
electrical measuring of small forces is very impressive for students.
to this they are attentive, if it is attempted to obtain a higher
sensitivity. The arrangement shown in fig. 4 can be recommended, if
the addition of forces is to be demonstrated. Wire D is put up
in horizontal position with a thread F. A load is applied to the
thread, how can be seen in fig. 4. In this case the students
see an unexpected high deflection of the measuring device .
quantitative examination of the facts seems appropriate now; it leads
to the addition
law of forces.
2.2 Introducing experiment to
the lever law
In fig. 5 you see a further
arrangement to measure little forces. The experiment indicated
is well suitable for motivating students on the treatment of
the lever law.
2.3 Registration of
In fig. 6 you
see an arrangement to observe vibration-diagrams. At
the wire D a screwing feather is hanging with a 100 g –
weight. If the feather is stretched, it exerts a force on wire D
proportional to the stretching. The signal corresponding to the
vibration is recorded by a computer with an ad-converter (look
at fig. 7).
3.4 Centre of
gravity-sentence and the theorem of momentum
According to fig. 8 wire D
carries the one end of a track, which is laid down
rotatably at the other end. A ball K rolling over the track causes an
increasing force on the wire D proportional to the distance from the
left end of the track. If the track is adjusted horizontally, then
the recording instrument draws a linear diagram. A linear diagramm
arises also, if two rolling glassballs collide on the track. In this
case the common centre of gravitiy is recorded. The linearity
indicates, that the centre of gravity is moving uniformly, if there
are no forces from the outside. Thus the centre of gravity
sentence is provided and the sentence of momentum can be concluded
With iron balls one gets
noticeable deviations of the ideal case because the forces exerted by
the track changing the angular momentums of the balls don’t
compensate themselves. It is caused by the friction between surfaces
of the iron balls in the moment of impact.
3.5 Uniformly accelerated
fig. 9 fig.
The diagram (parable) in
fig. 9 was recorded, when a ball rolled down the inclined track.
Initially the ball had been pushed upwards. This push was
assigned in a way, that it is exactly recognizable in the
diagram, when and where the downtrend starts. The straight line in
fig. 9 is a time-velocity diagram resulting from tangents,
which had been placed at the parable.
3.6 Measuring of momentum
Braking a rolling wagon by
a thread F connected with the wire D results in the curve in fig.10.
At the beginning of the experiment the thread is relaxed. The
momentum of the wagon can be destined by the computer.
Measuring of the centripetal force
At first the
wire D with an appended weight ist turned to the left, till the
weight reaches a point P, at which the wire is fastened. Then the
weight (100 g) is released and passes through the orbit with the
radius of r, indicated in fig.11 . During this event the
diagram in fig. 12 is recorded. At first it shows the gravity-force m
g (part A)
and then the force 3 ×
g (the peak)
, which is effective at the deepest point of the moving object. 3 ×
g is the sum
of the gravity-force m ×
the centripetal-force 2·m·g (the counter-force of the
the centripetal -force F = 2 ×
g with the
velocity of the moving object at the deepest point:
It is well
known, that the change of height is decisive for the
velocity. The same velocity is reached, if the object falls by r or
its height changes by r on the orbit.
Calculation of v at a falling
object: r = g/2·t2 , v = g ·
t → t = v/g
= 2 ·
g · r
= 2 ·
g · r ; F = 2×m×g
= 2 ·
m · g / (2 · g · r) = m/ r →
F = m · v2
a = F/m = v2/r
3.8 Stretching a lath of wood
At the right end of the lath the
wheatstone-bridge is fastened. Between the bridge and a little stick
on the left end of the lath the wire of constantan is tightened. If
two students pull at the ends of the lath, the pointer of a
measuring-instrument indicates, that the lath is stretched. This
experiment is suitable to clarify the law of action and reaction- the
piece of wood is stretched and reacts like a stretched coil spring.
3.9 Acceleration of the body
when bending the knees
In fig. 16 you see a woman on a
wooden board bending her knees. By the wheatstone-bridge fastened at
the underside of the board (look at fig. 17) the not visible bend of
the board will be measured. This bend is proportional to the
force F acting on the board.
F - m×
g = m ×
F = m×g
with an ad- converter will be taken, to measure this force. The
calibration happens by the weight of the body. Because Fweight
/ m = g is
= g + a will be measured by the board.
fig. 16 you see how the centre of gravity of the body is accelerated.
integration of the acceleration you get the velocity v.
The inserting of a new wire and the balancing of the bridge
A wire of constantan (diameter =
0.2 mm, approx. 90. 5 cm, 15 ohm) is inserted between the to clamps.
To avoid forces on the electrical contacts, the parts of the wire
near the contacts must be clamped at the circuit-board with the
round plate S.
If you intend to balance the
bridge, at first you must reduce the resistance of the wire. You
contact its two halves with one another. If the pointer
of the connected measuring-instrument doesn’t move (it remains
at the left or right edge of the scale), the resistance of the wire
is not sufficient. In this case the screw of the potentiometer
(rough) must be turned to the right, (otherwise it must be turned to
the left) until a deflection of the pointer occurs. Now
the pointer can be adjusted by the potentiometer (fine).
If it is impossible to
balance the bridge, there is a contact near the clamps between
the left and right end of the wire. In this case the voltage source
should be seperated immediately from the bridge, otherwise the
electrical power in some resistors would be too large.
The resitances of the resistors
A,B and C are dependent on the temperature ( 100 ppm/ K). Therefore
the wheatstone-bridge cannot be exposed to a wind.