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Question

The solid ball density is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is 0.5mm 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%, then the relative percentage error in the density is


A

0.9%

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B

2.4%

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C

3.1%

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D

4.2%

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Solution

The correct option is C

3.1%


Explanation for correct option

Step 1: Solve for the relative error in diameter of the ball

As we know that least count of a scale=pitchno.ofdivisions

LC of the scale=0.5500.01mm

Diameter of the ball will be=mainscalereading+(curvedsalereading×leastcount)

=2.5+(20×0.01)=2.5+0.2=2.7

Step 2: Solve for relative error in density

As density=MassVolume

volume of a ball=43πD23, where D=diameter of the ball

density=Mass43πD23

relative error in density is

ρρ=MM+3DD, where ρ=density,M=mass

Step 3 Solve for relative percentage error in density

relative percentage error in density is

ρρ×100=MM×100+3DD×100

MM×100=2D=0.01D=2.7 (given)

ρρ×100=2+3×0.012.7×100ρρ×100=2+109ρρ×100=289ρρ×100=3.1%

Hence, option C is correct.


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