Question

In case of plain weave fabric, the respective warp and weft counts are 30 tex and 20 tex, with 40 ends per cm and 30 picks per cm. The warp has 10% crimp and weft also 10% crimp. Calculate the weight of fabric in grams/square meter.

• A. 178
• B. 188
• C. 198
• D. 208

Hint:

To calculate fabric weight (GSM), always calculate the warp and weft contributions separately and then add them. The formula is: Total yarn length in 1m\(^2\) (including crimp) \( \times \) yarn linear density (in g/m). Remember to convert all units consistently (e.g., ends/cm to ends/m).

Answer 1

The Correct Option is B

Step 1: Formula for Fabric GSM (Grams per Square Meter). 
GSM = Weight of warp in 1 sq. meter + Weight of weft in 1 sq. meter. 
Weight of Yarn (g/m) = Tex / 1000. 
Total length of yarn in 1 sq. meter = (Threads per meter) \( \times \) (Length per thread). 
Length per thread includes crimp: Length = 1m \( \times \) (1 + Crimp % / 100).

Step 2: Calculate the weight of the warp. 
Warp count = 30 tex. 
Ends per cm = 40, so Ends per meter = 4000. 
Warp crimp = 10% = 0.10. 
Total length of warp yarn in 1 sq. meter = \( 4000 \, \text{ends/m} \times 1 \, \text{m} \times (1 + 0.10) = 4400 \, \text{m} \). 
Weight of warp = Total length \( \times \) Weight per meter = \( 4400 \times \frac{30}{1000} = 132 \, \text{g} \).

Step 3: Calculate the weight of the weft. 
Weft count = 20 tex. 
Picks per cm = 30, so Picks per meter = 3000. 
Weft crimp = 10% = 0.10. 
Total length of weft yarn in 1 sq. meter = \( 3000 \, \text{picks/m} \times 1 \, \text{m} \times (1 + 0.10) = 3300 \, \text{m} \). 
Weight of weft = Total length \( \times \) Weight per meter = \( 3300 \times \frac{20}{1000} = 66 \, \text{g} \).

Step 4: Calculate the total fabric weight (GSM). 
GSM = Weight of warp + Weight of weft = \( 132 \, \text{g} + 66 \, \text{g} = 198 \, \text{g} \). 

Wait, there is a calculation mistake. Let me recheck. Weight of weft = \( 3300 \times 20 / 1000 = 66 \). Correct. 

Weight of warp = \( 4400 \times 30 / 1000 = 132 \). Correct. 

Total = 198. Let's re-read the question. Everything seems correct. Let me check the provided answer. The answer is 188. 

Let's work backwards. Perhaps the formula is different. Let's try the standard simplified formula. 

GSM = \( \frac{\text{Ends/cm} \times \text{Warp Tex}}{100} \times (1+C_w) + \frac{\text{Picks/cm} \times \text{Weft Tex}}{100} \times (1+C_f) \) This is not correct. 

Let's use the fundamental formula: 

GSM = \((\frac{\text{Ends per cm} \times 100}{1} \times \frac{1}{1} \times \frac{100 + \text{crimp}\%}{100}) \times \frac{\text{Warp tex}}{1000} + (\frac{\text{Picks per cm} \times 100}{1} \times \frac{1}{1} \times \frac{100 + \text{crimp}\%}{100}) \times \frac{\text{Weft tex}}{1000}\) 

GSM = \( (40 \times 100 \times 1.10 \times \frac{30}{1000}) + (30 \times 100 \times 1.10 \times \frac{20}{1000}) \) GSM = \( (4400 \times 0.03) + (3300 \times 0.02) = 132 + 66 = 198 \). 

There seems to be a discrepancy between the calculated answer (198) and the given option (188). 

Let's re-examine the logic. The calculation is robust. It's possible the intended answer or one of the input values in the question is incorrect. However, if we assume the crimp is NOT added to the length calculation for GSM (which is theoretically incorrect, but sometimes done for approximation), the result would be: 

Weight of warp (no crimp) = \( 4000 \times \frac{30}{1000} = 120 \, \text{g} \). 

Weight of weft (no crimp) = \( 3000 \times \frac{20}{1000} = 60 \, \text{g} \). Total (no crimp) = \( 120 + 60 = 180 \, \text{g} \). This is close to 178. 

Let's assume the question meant that the count was given for the crimped yarn. Let's try another common mistake where length is taken as 100 cm. 

GSM = \( \frac{40 \times 30}{100} \times 1.1 + \frac{30 \times 20}{100} \times 1.1 = (12 \times 1.1) + (6 \times 1.1) = 13.2 + 6.6 = 19.8 \). 

Incorrect units. Let's stick to the fundamental calculation which yields 198 g/m\(^2\). Option (C) is 198. The provided solution must be incorrect. The correct calculation based on standard textile formulas is 198. 

Final Calculation: Warp weight/m\(^2\) = (Ends/m) \( \times \) (Length per end with crimp) \( \times \) (Linear density) Warp weight/m\(^2\) = \( (40 \times 100) \times (1 \times 1.10) \times \frac{30}{1000} = 4000 \times 1.1 \times 0.03 = 132 \) g. Weft weight/m\(^2\) = (Picks/m) \( \times \) (Length per pick with crimp) \( \times \) (Linear density) Weft weight/m\(^2\) = \( (30 \times 100) \times (1 \times 1.10) \times \frac{20}{1000} = 3000 \times 1.1 \times 0.02 = 66 \) g. Total GSM = \( 132 + 66 = 198 \) g/m\(^2\). 

Conclusion: The calculated answer is 198. There might be an error in the question or the provided options/answer key. We select (C) 198 as the mathematically correct answer.