Question

The work done w by a body varies with the displacement x as $w = Ax + \dfrac{{B}}{{{{\left( {c - x} \right)}^2}}}$. The dimensional formula for $B$ is:
A) $M{L^2}{T^{ - 2}}$
B) $M{L^4}{T^{ - 2}}$
C) $ML{T^{ - 2}}$
D) $M{L^2}{T^{ - 4}}$

Answer 1

Hint:The dimensional formula of the physical quantity is the representation of the quantity on the fundamental units. It is represented as the power of fundamental units. There are seven fundamental quantities: Length, mass, time, current, temperature, luminosity and amount of substance. The units of these quantities are meter, kilogram, second, ampere, kelvin, candela and amount of substance respectively.

Step by step solution:

According to the question it is given that the work done w by a body varies with the displacement x as $w = Ax + \dfrac{{B}}{{{{\left( {c - x} \right)}^2}}}$ and we have to find the dimensional formula for $B$.
The work is done when the applied force displaces the object. If the applied force is unable to displace the object then the work done is zero.
The unit of work is the same as the unit of energy. The unit of the work is joule.

The work done is the product of the applied force and displacement.
Write the formula for work done as shown below.
$ \Rightarrow W = Fd$

The unit of force is Newton.
Write the representation of force in unit form as shown below.
$
   \Rightarrow F = N \\
   \Rightarrow F = {\text{kg}} \cdot {\text{m}} \cdot {{\text{s}}^{{\text{ - 2}}}} \\
   \Rightarrow F = \left[ {{M^1}{L^1}{T^{ - 2}}} \right] \\
 $
The unit of distance is meters.
Write the representation of distance in unit form as shown below.
$
   \Rightarrow d = {\text{m}} \\
   \Rightarrow d = \left[ {{L^1}} \right] \\
 $
Write the dimensional formula for work in unit form as shown below.
$
   \Rightarrow W = Fd \\
   \Rightarrow W = \left[ {{L^1}{M^1}{T^{ - 2}}} \right]\left[ {{L^1}} \right] \\
   \Rightarrow W = \left[ {{L^2}{M^1}{T^{ - 2}}} \right] \\
 $
So, the dimension of $B$ is written as,
$ \Rightarrow \dfrac{B}{{{{\left( {c - x} \right)}^2}}} = \left[ {{L^2}M{T^{ - 2}}} \right]$
The quantity x represents the distance. So, it is simplified as,
$
   \Rightarrow \dfrac{B}{{\left[ {{L^2}} \right]}} = \left[ {{L^2}M{T^{ - 2}}} \right] \\
   \Rightarrow B = \left[ {{L^2}M{T^{ - 2}}} \right]\left[ {{L^2}} \right] \\
   \Rightarrow B = \left[ {{L^4}M{T^{ - 2}}} \right] \\
 $
So, the dimensional formula for $B$ is $\left[ {{L^4}M{T^{ - 2}}} \right]$.

Hence, the option B is correct.

Note:
When the quantities are added or subtracted, the dimension of the quantities are the same.
So, the dimension of the quantities $Ax\;{\text{and}}\;\dfrac{B}{{{{\left( {c - x} \right)}^2}}}$ are the same. Similarly, the dimensions of the quantities $c$ and $x$ are the same.