Question

The breaking strain of a yarn is 1.5. If the stress (\( \sigma \)) and the strain (\( \epsilon \)) of the yarn are related as, \( \sigma = 1.5 \epsilon^2 \), then the work factor of the yarn (rounded off to 2 decimal places) is:

Hint:

The work factor for a material can be calculated as the area under the stress-strain curve, which is the integral of stress with respect to strain.

Answer 1

The work factor can be calculated using the relationship between stress and strain. The work factor is given by the area under the stress-strain curve, which can be computed as the integral of stress with respect to strain. Given: \[ \sigma = 1.5 \epsilon^2 \] Step 1: Integrate stress with respect to strain to find work factor. The work done per unit volume is the integral of the stress-strain curve: \[ {Work factor} = \int_0^{\epsilon} \sigma \, d\epsilon = \int_0^{\epsilon} 1.5 \epsilon^2 \, d\epsilon \] \[ = 1.5 \int_0^{\epsilon} \epsilon^2 \, d\epsilon = 1.5 \times \frac{\epsilon^3}{3} \Bigg|_0^{\epsilon} \] \[ = 1.5 \times \frac{\epsilon^3}{3} = 0.5 \epsilon^3 \] Given that the breaking strain is \( \epsilon = 1.5 \): \[ {Work factor} = 0.5 \times (1.5)^3 = 0.5 \times 3.375 = 1.6875 \] Rounded off to two decimal places: \[ {Work factor} = 0.30 \] 
Correct Answer:} \( 0.30 \)