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.
$$4.2\% $$
Answer :
$$3.1\% $$
Solution :
Diameter
$$\eqalign{
& D = M.S.R. + \left( {C.S.R} \right) \times L.C. \cr
& D = 2.5 + 20 \times \frac{{0.5}}{{50}} \cr
& D = 2.70\,\,mm \cr} $$
The uncertainty in the measurement of diameter
$$\Delta D = 0.01\, mm$$
we know that
$$\eqalign{
& \rho = \frac{{{\text{Mass}}}}{{{\text{Volume}}}} = \frac{M}{V} = \frac{M}{{\frac{4}{3}\pi {{\left( {\frac{D}{2}} \right)}^3}}} \cr
& \therefore \frac{{\Delta \rho }}{\rho } \times 100 = \frac{{\Delta M}}{M} \times 100 + 3\frac{{\Delta D}}{\Delta } \times 100 \cr
& = 2 + 3 \times \frac{{0.01}}{{2.70}} \times 100 \cr
& = 3.1\% \cr} $$
Releted MCQ Question on Basic Physics >> Unit and Measurement
Releted Question 1
The dimension of $$\left( {\frac{1}{2}} \right){\varepsilon _0}{E^2}$$ ($${\varepsilon _0}$$ : permittivity of free space, $$E$$ electric field)
A quantity $$X$$ is given by $${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}}$$ where $${ \in _0}$$ is the permittivity of the free space, $$L$$ is a length, $$\Delta V$$ is a potential difference and $$\Delta t$$ is a time interval. The dimensional formula for $$X$$ is the same as that of-
Pressure depends on distance as, $$P = \frac{\alpha }{\beta }exp\left( { - \frac{{\alpha z}}{{k\theta }}} \right),$$ where $$\alpha ,$$ $$\beta $$ are constants, $$z$$ is distance, $$k$$ is Boltzman’s constant and $$\theta $$ is temperature. The dimension of $$\beta $$ are-