Question

The quantities A and B are related by the relation, $m=\dfrac{A}{B}$, when m is the linear density and A is the force. The dimensions of B are of
A. Pressure
B. latent heat
C. Work
D. None of the above

Answer 1

Hint: Rearrange the given equation as $B=\dfrac{A}{m}$ and calculate the dimensional formula of quantity B with the help of the given information. Then find out which of the quantities in the given options has the same dimensional formula. If the dimensional formula is equal then the dimensions of the two quantities are same.

Formula used: $L=\dfrac{Q}{m}$

Complete step by step answer:
Let us first calculate the dimensional formula of the quantity B and see which of the options have the same dimensional formula.
We will make the use of the given relation between the quantities A and B. It is given that $m=\dfrac{A}{B}$ …. (i)
We can write equation (i) as
$B=\dfrac{A}{m}$ ….. (ii)
Therefore, the dimensional formula of quantity B will be the ratio of the dimensional formula of quantity A to the ratio of the dimensional formula of quantity m.
i.e. $\left[ B \right]=\dfrac{\left[ A \right]}{\left[ m \right]}$ ….. (iii).
It is given that quantity A is a force. We know that the dimensional formula of force is $\left[ ML{{T}^{-2}} \right]$. This means that the dimensional formula of A is $\left[ A \right]=\left[ ML{{T}^{-2}} \right]$.
It is given that quantity m is linear density. Let us assume that it is linear mass density. Linear mass density is defined as mass per unit length. Hence, the dimensional formula of linear mass density is $\left[ M{{L}^{-1}} \right]$. This means that the dimensional formula of m is $\left[ m \right]=\left[ M{{L}^{-1}} \right]$.
Substitute the dimensional formulas of the quantities A and m in equation (iii).
Therefore, we get that,
$\left[ B \right]=\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ M{{L}^{-1}} \right]}=\left[ \dfrac{ML{{T}^{-2}}}{M{{L}^{-1}}} \right]=\left[ {{L}^{2}}{{T}^{-2}} \right]$.
This means that the dimensional formula of quantity B is $\left[ {{L}^{2}}{{T}^{-2}} \right]$.
Latent heat of a substance is the amount of heat energy required to change the phase of one unit mass of the substance. Hence, latent heat (L) has the dimension same as the ratio of the dimension of heat energy (Q) to the dimension of mass (m).
This means that the dimensional formula of latent heat is $\left[ L \right]=\dfrac{\left[ Q \right]}{\left[ m \right]}$.
We know that the dimensional formulas of energy and mass are $\left[ M{{L}^{2}}{{T}^{-2}} \right]$ and [M] respectively.
This gives us that $\left[ L \right]=\dfrac{\left[ Q \right]}{\left[ m \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ M \right]}=\left[ {{L}^{2}}{{T}^{-2}} \right]$.
This means that the dimensional formulas of latent heat and quantity B are the same.

So, the correct answer is “Option B”.

Note: Let us see the dimensional formulas of the incorrect options.
Pressure is force (F) per unit area (A). Therefore, dimensional formula of pressure (P) is $\left[ P \right]=\dfrac{\left[ F \right]}{\left[ A \right]}=\dfrac{\left[ ML{{T}^{-2}} \right]}{\left[ {{L}^{2}} \right]}=\left[ M{{L}^{-1}}{{T}^{-2}} \right]$.
Work done is the product of force (F) and displacement (d). Therefore, the dimensional formula of work (W) is $\left[ W \right]=\left[ F \right]\left[ d \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
In these types of questions, one can also check for the units of the quantities. If the units of the quantities are the same, then the quantities have the same dimensions.