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 0.5mm there are 50 divisions on the circular scale. The reading on the main scale is 2.5mm 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(Approximate the answer to the nearest integer.)

Question

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 0.5mm there are 50 divisions on the circular scale. The reading on the main scale is 2.5mm 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(Approximate the answer to the nearest integer.) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Least count =(0 . 5/50)mm diameter of ball =2.5mm+20× (0 . 5/50) =2.7mm ρ =(M/V)=(M/(4)3 π R3) lnρ =lnM-ln((4/3) π R3) (Δ ρ /ρ )× 100=(Δ M/M)× 100+3(Δ R/R)× 100 =3.1 %

Least count =500.5​mm diameter of ball =2.5mm+20×500.5​ =2.7mm ρ=VM​=34​πR3M​ lnρ=lnM−ln(34​πR3) ρΔρ​×100=MΔM​×100+3RΔR​×100 =3.1%

Least count $= \frac{0.5}{50} mm$
diameter of ball $= 2.5 mm + 20 \times \frac{0.5}{50}$
$= 2.7 mm$
$ρ = \frac{M}{V} = \frac{M}{\frac{4}{3} π R ^{3}}$
$l n ρ = l n M - l n (\frac{4}{3} π R^{3})$
$\frac{Δ ρ}{ρ} \times 100 = \frac{Δ M}{M} \times 100 + 3 \frac{Δ R}{R} \times 100$
$= 3.1%$