Question

In the force transducer shown in Figure (a), four identical strain gauges S1, S2, S3, and S4 are mounted on a cantilever at equal distance from its base. S1 and S2 are mounted on the top surface and S3 and S4 are mounted on the bottom surface, as shown in the Figure (a). These strain gauges are to be connected to form a Wheatstone bridge consisting of four arms A, B, C, and D, as shown in the Figure (b). From the following options, the correct order to maximize the measurement sensitivity is

 

• A. A → S1, B → S2, C → S4, D → S3
• B. A → S1, B → S4, C → S3, D → S2
• C. A → S1, B → S2, C → S3, D → S4
• D. A → S1, B → S4, C → S2, D → S3

Hint:

Maximum sensitivity of a Wheatstone bridge with strain gauges is achieved when gauges experiencing opposite strains are placed in adjacent arms of the bridge. This configuration leads to the largest change in the output voltage for a given applied force.

Answer 1

The Correct Option is B

Step 1: Understand the effect of force on the cantilever and strain gauges.
When a downward force is applied to the cantilever beam, the top surface experiences tensile strain (increase in resistance, \(+\Delta R\)), and the bottom surface experiences compressive strain (decrease in resistance, \(-\Delta R\)).

Step 2: Recall the output voltage of a Wheatstone bridge.
The output voltage \(V_o\) of a Wheatstone bridge is proportional to the difference in the ratios of the resistances of the arms.

Step 3: Analyze Option (B): A → S1 (\(R + \Delta R\)), B → S4 (\(R - \Delta R\)), C → S3 (\(R - \Delta R\)), D → S2 (\(R + \Delta R\)).
In this configuration, the bridge arms are:
A: \(R + \Delta R\)
B: \(R - \Delta R\)
C: \(R - \Delta R\)
D: \(R + \Delta R\)

The output voltage is proportional to \[ \frac{B}{A+B} - \frac{C}{D+C} = \frac{R - \Delta R}{(R + \Delta R) + (R - \Delta R)} - \frac{R - \Delta R}{(R + \Delta R) + (R - \Delta R)}. \]
Let's reconsider the arrangement for maximum sensitivity. We want strain gauges with opposite signs of \(\Delta R\) in adjacent arms.

In Option (B):
A (S1): Tension (\(+\Delta R\))
B (S4): Compression (\(-\Delta R\))
C (S3): Compression (\(-\Delta R\))
D (S2): Tension (\(+\Delta R\))

This arrangement places tensile and compressive strain gauges in adjacent arms, which maximizes the unbalance of the bridge and hence the sensitivity.