Page 402 - Physics Coursebook 2015 (A level)

Cambridge International A Level Physics
WORKED EXAMPLE
390
they happen. This provides important information when an aircraft is tested.
In Chapter 11, we saw that a wire of length L, cross- sectional area A and resistivity ρ has a resistance R given by:
R = ρL A
What happens to the resistance when the wire is stretched depends on the construction of the strain gauge. The simplest approximation is to assume that the cross- sectional area of the wire remains unchanged.
If the wire increases in length by a small amount δL and the cross-sectional area A is unchanged, then the resistance of the wire increases by δR, where:
R + δR = ρ(L + δL) A
2
The wire in a strain gauge when unstretched has length 10.00 cm and resistance 120.0 Ω. When the wire is stretched by 0.10 cm, the resistance becomes 122.4 Ω. The strain gauge is connected in the circuit shown in Figure 25.8. What is the change in length of the wire in the strain gauge when the output voltage is5.06V?
10.0 V d.c.
120.0 Ω
strain gauge
Subtracting the two equations gives: δR = ρδL
output voltage
A ρL
If the expression is divided by R = A , we get:
Figure 25.8 Circuit for Worked example 2.
Step1 Determinetheresistanceofthestrain gauge Rs by considering the two resistors as a potential divider. The 10 V supply is split in the ratio 5.06V:4.94Vandso:
Rs = 5.06 × 120 = 122.9 Ω 4.94
δR = δL RL
When a wire is stretched not only does its length increase; at the same time, its area decreases (because its volume remains roughly constant). This also increases the wire’s resistance. The increase in resistance can be shown to be the same as that caused by the increase in length so that, for small changes:
δR=2ρδL or δR=2δL ARL
In either case, the change in resistance is directly proportional to the increase in length, δR ∝ δL.
The change in resistance is likely to be very small but with a suitable series resistor and a cell used as a sensor, the change in length can produce a measurable change in output voltage.
QUESTION
6 Using the data given in Worked example 2:
a calculate the increase in length of the wire
when the output voltage is 5.1 V
b calculate the strain in the wire when the output voltage is 5.1 V
c calculate the output voltage when the wire is stretched by 0.05 cm.
Alternatively, apply the potential divider formula Vout= R2 ×Vin
R1×R2
120 × 10.0
4.94 =
It is easier to use 4.94 V, the voltage across the 120 Ω
120+Rs
resistor. This gives a value for Rs of 122.9 Ω.
Step2 Theresistanceofthewirehasincreasedby 2.9 Ω. Since δR ∝ δL, and an increase in length of 0.10 cm increases the resistance by 2.4 Ω, the length has increased by:
2.9 ×0.10=0.12cm 2.4
If the strain gauge is glued to a metal support, the strain in the metal support is the same as the strain
in the gauge, in this case 0.12 = 1.2%. Thus the strain 10
gauge can easily measure strain.