The dimensions of a cone are measured using a scale with a least count of $2 mm$. The diameter of the base and the height are both measured to be $20.0 cm$. The maximum percentage error in the determination of the volume is. . . . .

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z=x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$, respectively, then

$z \pm \Delta z=\frac{x \pm \Delta x}{y \pm \Delta y}=\frac{x}{y}\left(1 \pm \frac{\Delta x}{x}\right)\left(1 \pm \frac{\Delta y}{y}\right)^{-1} .$

The series expansion for $\left(1 \pm \frac{\Delta y}{y}\right)^{-1}$, to first power in $\Delta y / y$, is $1 \mp(\Delta y / y)$. The relative errors in independent variables are always added. So the error in $z$ will be $\Delta z=z\left(\frac{\Delta x}{x}+\frac{\Delta y}{y}\right)$.

The above derivation makes the assumption that $\Delta x / x \ll<1, \Delta y / y \ll<1$. Therefore, the higher powers of these quantities are neglected.

($1$) Consider the ratio $r =\frac{(1- a )}{(1+ a )}$ to be determined by measuring a dimensionless quantity a.

If the error in the measurement of $a$ is $\Delta a (\Delta a / a \ll<1)$, then what is the error $\Delta r$ in

$(A)$ $\frac{\Delta a }{(1+ a )^2}$  $(B)$ $\frac{2 \Delta a }{(1+ a )^2}$  $(C)$ $\frac{2 \Delta a}{\left(1-a^2\right)}$  $(D)$ $\frac{2 a \Delta a}{\left(1-a^2\right)}$

($2$) In an experiment the initial number of radioactive nuclei is $3000$ . It is found that $1000 \pm$ 40 nuclei decayed in the first $1.0 s$. For $|x|<1$, In $(1+x)=x$ up to first power in $x$. The error $\Delta \lambda$, in the determination of the decay constant $\lambda$, in $s ^{-1}$, is

$(A) 0.04$  $(B) 0.03$  $(C) 0.02$  $(D) 0.01$

Give the answer or quetion ($1$) and ($2$)

The least count of stop watch is $\frac{1}{5}\,second$. The time of $20$ oscillations of pendulum is measured to be $25\,seconds$. Then percentage error in the measurement of time will be.......... $\%$

The mass of the body is $10.000\,g$ and its volume is $10.00\,cm^3$. If the measured values are expressed upto the correct significant figures, the maximum error in the measurement of density is

If a copper wire is stretched to make its radius decrease by $0.1\%$ , then percentage increase in resistance is approximately .......... $\%$

Following observations were taken with a vernier callipers while measuring the length of a cylinder

$3.29 \,cm, 3.28\, cm, 3.29 \,cm, 3.31 \,cm,$ $ 3.28\, cm, 3.27 \,cm, 3.29 \,cm, 3.30\, cm$

Then find Absolute error in forth and eighth observation