Question 32: Units , Dimensions and Error Analysis

Question

Match List - I with List - II.

List – I
List – II

A.  Coefficient of viscosity
      B.  Surface tension
      C.  Pressure
      D.  Surface energy

I.  [ML−1T−2]
      II.  [ML2T−2]
      III.  [ML0T−2]
      IV.  [ML−1T−1]

Choose the correct answer from the options given below :

Accepted Answer

From Newton's law of viscosity, the force is $\mathrm{F}=\eta \mathrm{A}\left(\frac{\Delta \mathrm{v}}{\Delta \mathrm{x}}ight)$, where A is area and $\frac{\Delta \mathrm{v}}{\Delta \mathrm{x}}$ is the velocity gradient.
$$ \eta=\frac{\mathrm{F}}{\mathrm{~A} \cdot\left(\frac{\Delta \mathrm{v}}{\Delta \mathrm{x}}ight)} $$
$$\Rightarrow $$ $$ [\eta]=\frac{\left[\mathrm{MLT}^{-2}ight]}{\left[\mathrm{L}^2ight] \cdot\left(\frac{\left[\mathrm{LT}^{-1}ight]}{[\mathrm{L}]}ight)}=\frac{\left[\mathrm{MLT}^{-2}ight]}{\left[\mathrm{L}^2ight] \cdot\left[\mathrm{T}^{-1}ight]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}ight] $$
$$ \mathrm{A} ightarrow \mathrm{IV} $$
Surface tension is defined as the force acting per unit length.
$$ \mathrm{T}=\frac{	ext { Force }}{	ext { Length }} $$
$$ [\mathrm{T}]=\frac{\left[\mathrm{MLT}^{-2}ight]}{[\mathrm{L}]}=\left[\mathrm{ML}^0 \mathrm{~T}^{-2}ight] $$
B $$	o $$ III
Pressure is defined as force acting per unit area.
$$ \mathrm{P}=\frac{	ext { Force }}{	ext { Area }} $$
$$ [\mathrm{P}]=\frac{\left[\mathrm{MLT}^{-2}ight]}{\left[\mathrm{L}^2ight]}=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}ight] $$
$$ C ightarrow I $$
Surface energy $\left(\mathrm{U}_{\mathrm{s}}ight)$ is defined as the work done by surface tension in changing the surface area
$$ \mathrm{U}_{\mathrm{s}}=\mathrm{T} \cdot(\Delta \mathrm{~A}) $$
$$\Rightarrow $$ $\left[\mathrm{U}_{\mathrm{S}}ight]=[$ Surface Tension $] \cdot[$ Area $]$
$$\Rightarrow $$ $$ \left[\mathrm{U}_{\mathrm{S}}ight]=\left[\mathrm{ML}^0 \mathrm{~T}^{-2}ight] \cdot\left[\left[\mathrm{M}^0 \mathrm{~L}^2 \mathrm{~T}^0ight]ight]=\left[\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]ight] $$
$$ \mathrm{D} ightarrow \mathrm{II} $$
$$ \begin{aligned} &\begin{array}{|l|l|} \hline 	ext { List - I } & 	ext { List - II } \ \hline 	ext { A. Coefficient of viscosity } & 	ext { IV. }\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}ight] \ \hline 	ext { B. Surface tension } & 	ext { III. }\left[\mathrm{ML}^0 \mathrm{~T}^{-2}ight] \ \hline 	ext { C. Pressure } & 	ext { I. }\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}ight] \ \hline 	ext { D. Surface energy } & 	ext { II. }\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight] \ \hline \end{array}\ &	ext { Hence, the correct option is (B). } \end{aligned} $$

Answer

A-I, B-II, C-IV, D-III

Answer

A-IV, B-III, C-I, D-II

Answer

A-IV, B-I, C-II, D-III

Answer

A-I, B-III, C-II, D-IV

Units , Dimensions and Error Analysis

Match List - I with List - II. List – I List – II A. Coefficient of viscosity B. Surface tension C. Pressure D. Surface energy I. [ML−1T−2] II. [ML2T−2] III. [ML0T−2] IV. [ML−1T−1] Choose the correct answer from the options given below :

Choose the correct answer: