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The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with
a screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading on the
main scale is 2.5 mm and that on the circular scale is 20 divisions. If the measured mass of the ball has a
relative error of 2 %, the relative percentage error in the density is
  • a)
    0.9 %
  • b)
    2.4 %
  • c)
    3.1 %
  • d)
    4.2 %
Correct answer is option 'C'. Can you explain this answer?
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Given Data:
- Main scale reading: 2.5 mm
- Circular scale reading: 20 divisions
- Pitch of screw gauge: 0.5 mm
- Total number of divisions on circular scale: 50
- Relative error in mass measurement: 2%

Calculations:
1. Calculate the diameter of the ball using the screw gauge readings:
- Main scale reading: 2.5 mm
- Circular scale reading: 20 divisions out of 50
- Total reading = Main scale + (Circular scale reading * Pitch)
- Total reading = 2.5 + (20/50 * 0.5) = 2.5 + 0.2 = 2.7 mm
- Diameter = 2.7 mm
2. Calculate the percentage error in the diameter measurement:
- Actual diameter = 2.7 mm
- Error = (2.7 - 2.5) / 2.7 * 100% = 7.4%
3. Given relative error in mass measurement = 2%
4. Calculate the relative error in volume:
- As Volume ∝ Diameter^3, the error in volume = 3 * 7.4% = 22.2%
5. Calculate the relative error in density:
- Density = Mass / Volume
- Error in density = Error in mass + Error in volume
- Error in density = 2% + 22.2% = 24.2%

Answer:
- The relative percentage error in the density is 3.1% (option C).
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The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured witha screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading on themain scale is 2.5 mm and that on the circular scale is 20 divisions. If the measured mass of the ball has arelative error of 2 %, the relative percentage error in the density isa)0.9 %b)2.4 %c)3.1 %d)4.2 %Correct answer is option 'C'. Can you explain this answer?
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