Question

A nail of mass ‘${{m}_{1}}$’ kg is being hammered by a hammer of mass ‘${{m}_{2}}$’ kg with a velocity of ‘v’ m/s such that the nail drives by ‘s ’m into a wall. Find the average resistance offered by the wall to the penetration of nails.
(A) $[\dfrac{{{m}_{2}}v}{({{m}_{1}}+{{m}_{2}})2s}]$N
(B) $[\dfrac{{{m}_{2}}{{v}^{2}}}{({{m}_{1}}+{{m}_{2}})2s}]$ N
(C) $[\dfrac{{{m}_{2}}^{2}v}{({{m}_{1}}+{{m}_{2}})2s}]N$
(D) $[\dfrac{m_{2}^{2}{{v}^{2}}}{({{m}_{1}}+{{m}_{2}})2s}]N$

Answer 1

Hint: We can solve this question by using Newton’s law to find the resistance offered by the wall. But by analyzing the given options, we can deduce an easier way to find out the resistance without going into many calculations. This is the method of dimensional analysis with which the answer can be derived only with a couple of steps.

Complete answer:
By using the concept of dimensional analysis on the given options, we solve the question to get the correct answer.
Using dimensional analysis on option (A), we have
$\begin{align}
  & \dfrac{M\times (L{{T}^{-1}})}{M\times L} \\
 & ={{T}^{-1}} \\
\end{align}$ ……….(1)
From option (B), we get
$\begin{align}
  & \dfrac{M{{(L{{T}^{-1}})}^{2}}}{ML} \\
 & =L{{T}^{-2}} \\
\end{align}$ ………… (2)
From option (C), we have,
$\begin{align}
  & \dfrac{{{M}^{2}}L{{T}^{-1}}}{ML} \\
 & =M{{T}^{-1}} \\
\end{align}$ …………..(3)
From option (D), we have
$\begin{align}
  & \dfrac{{{M}^{2}}{{(L{{T}^{-1}})}^{2}}}{ML} \\
 & =ML{{T}^{-2}} \\
\end{align}$ …………….(4)
The average resistance offered will be in the form of a force so the average resistance will have the same units as the units of force given by
Force=mass$\times$acceleration
Unit of force=unit of mass$\times$unit of acceleration
$\Rightarrow [F]=[M]\times [L{{T}^{-2}}]$
$\Rightarrow [F]=[ML{{T}^{-2}}]$
This unit of force is only obtained in option D from (4). Therefore, the correct option is option (D) since it has the same unit as that of resistance.

Thus, (D) $[\dfrac{m_{2}^{2}{{v}^{2}}}{({{m}_{1}}+{{m}_{2}})2s}]N$ is the correct answer for the resistance offered by the wall.

Additional Information:
Dimensional analysis is the key to solve many problems. In fact at the very first instance of studying a problem, the dimensional analysis must be checked. It helps to eliminate options or even helps us to find the correct answer without much effort or time.

Note:
This problem can also be attempted by making use of Newton’s law of motion, the 3rd law of Newton in particular which states that every action has an equal and opposite reaction. The action force here is the force of the nail on the wall and the reaction force is the resistance provided by the wall. But dimensional analysis saves us time and effort so we go with that approach instead.