A 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
How does the resistance of a wire by stretching, by bending and warming change ? This 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 crosssection can be examined and the specific resistance can be introduced.
1. Examining the change of resistance by warming, stretching and bending
The change 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 Ohmresistors A, B, C and D. D, the measuringelement is a constantanwire (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 measuringsignal .


The inserting of a new wire and the balancing of the bridge.

fig. 1
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.
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 equal. For 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 10^{o} 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.
Taking a 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 manganinconstantan by hand causes a strong deflection of the measuring device. The deflection then decreases exponentially.
fig. 3
2. Experiments 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.
2.1 Introduction experiment: Addition of forces
The demonstration of a wire of constantan as a measuringelement for electrical measuring of small forces is very impressive for students. According 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 . A quantitative examination of the facts seems appropriate now; it leads to the addition law of forces.
fig. 4 fig. 5
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 oscillations
fig. 6 fig. 7
In fig. 6 you see an arrangement to observe vibrationdiagrams. 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 adconverter (look at fig. 7).
3.4 Centre of gravitysentence and the theorem of momentum
fig. 8
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 from this.
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 motion
fig. 9 fig. 10
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 timevelocity 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.
3.7 Measuring of the centripetal force
fig. 11 fig. 12
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 gravityforce m × g (part A) and then the force 3 × m× g (the peak) , which is effective at the deepest point of the moving object. 3 × m× g is the sum of the gravityforce m × g and the centripetalforce 2·m·g (the counterforce of the centripetalforce).
We compare the centripetal force F = 2 × m× 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·t^{2} , v = g · t → t = v/g
→ r = g/2·(v/g)^{2 } → v^{2} = 2 · g · r
v^{2} = 2 · g · r ; F = 2×m×g → F/v^{2} = 2 · m · g / (2 · g · r) = m/ r → F = m · v^{2} / r
→ Centripetalacceleration: a = F/m = v^{2}/r
3.8 Stretching a lath of wood
fig. 13
At the right end of the lath the wheatstonebridge 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 measuringinstrument 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 wheatstonebridge 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 × a ® F = m×g + m×a
A computer with an ad converter will be taken, to measure this force. The calibration happens by the weight of the body. Because F_{weight} / m = g is assigned, F/m = g + a will be measured by the board. In fig. 16 you see how the centre of gravity of the body is accelerated. By integration of the acceleration you get the velocity v.
fig. 15 fig. 16
fig. 17
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 circuitboard 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 measuringinstrument 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).
Attention !
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.
Essential note:
The resitances of the resistors A,B and C are dependent on the temperature ( 100 ppm/ K). Therefore the wheatstonebridge cannot be exposed to a wind.