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%

Given Screw gauge readings
Pitch = 0.5 mm
Circular scale division = 50
Main scale reading = 2.5 mm
Least count = 0.5/50 = 0.01 mm
Circular scale division reading = 20 divisions
Relative error = 2%
Diameter = Main scale reading + (Circular scale reading × Least count)
Diameter = 2.5 mm + (20 × 0.01 mm) = 2.7 mm = 0.27 cm
Radius = Diameter/2 = 0.27 cm/2 = 0.135 cm
Volume = (4/3)πr³ = (4/3) × 3.14 × (0.135 cm)³ = 0.10306 cm³
Let the mass of the ball be m grams.
Relative error in mass = 2%
Error in mass = 2% of m = 0.02m
Density = mass/volume = m/0.10306 g/cm³
Error in density = (error in mass/mass) + (3 × error in radius/radius) (Since volume is proportional to r³)
Assuming the error in radius is negligible compared to the error in mass:
Error in density ≈ error in mass/mass = 0.02 = 2%
Therefore, the relative error in density is approximately 2%.