Question

The mass of a body is 10.000gm and its volume is $10.00c{{m}^{3}}$. If the measured values are expressed up to correct significant figures, the maximum error in the measurement of the density is:
$\begin{align}
  & a)0.0012gm/c{{m}^{3}} \\
 & b)0.0011gm/c{{m}^{3}} \\
 & c)0.0001gm/c{{m}^{3}} \\
 & d)0.021gm/c{{m}^{3}} \\
\end{align}$

Answer 1

Hint: It is given in the question to us that the above values measured for the mass and the volume of the body are expressed up to correct significant figures. From this we can imply that the value of the quantity after the particular figures after the decimal point are not considered. Hence we will determine the maximum error in both the mass and the volume of the body. Since density is given as the ratio of mass to that of volume, using the expression of error in mass and volume we will determine the error in density of the body.

Formula used:
$\dfrac{\Delta d}{d}=\dfrac{\Delta m}{m}+\dfrac{\Delta V}{V}$

Complete step-by-step answer:
The values measured up to the correct significant figures of volume (V)is $10.00c{{m}^{3}}$ and that of mass (m)is 10.000gm. hence from this we can conclude the maximum possible error in mass $\Delta m$and volume $\Delta V$ is equal to,
$\Delta m=0.001gm\text{ and }\Delta V=0.01c{{m}^{3}}$
The density of a substance is given as the ratio of its mass to that of its volume. Hence the density(d) of the above body is,
$\begin{align}
  & d=\dfrac{m}{V} \\
 & \Rightarrow d=\dfrac{10.000gm}{10.00c{{m}^{3}}}=1gm/c{{m}^{3}} \\
\end{align}$
The error in density of the body i.e. $\Delta d$ in terms of the error in its volume and mass is given by,
$\dfrac{\Delta d}{d}=\dfrac{\Delta m}{m}+\dfrac{\Delta V}{V}$
We know the density, volume and mass of the substance as well we now know the error in the mass and the volume of the body. Therefore the error in the density of the substance is equal to,
$\begin{align}
  & \dfrac{\Delta d}{d}=\dfrac{\Delta m}{m}+\dfrac{\Delta V}{V} \\
 & \Rightarrow \dfrac{\Delta d}{1}=\dfrac{0.001}{10.000}+\dfrac{0.01}{10.00} \\
 & \Rightarrow \Delta d=\dfrac{0.011}{10.000}=\dfrac{11}{10000}=0.0011gm/c{{m}^{3}} \\
\end{align}$
Hence the correct answer of the above question is option b.

So, the correct answer is “Option b”.

Note: Error in the measurement is defined as the difference between the actual reading and the measured value of the quantity. The above expression for error in a physical quantity is the same for the error in physical quantity which comprises the product of other quantities. It is to be noted that the error in the physical quantity always adds up.