Question

The quantities A and B are related by the relation $\dfrac{A}{B} = m$ , where m is the linear mass density and A is the force, the dimensions of B will be:
(A) Same as that of pressure
(B) Same as that of work
(C) Same as that of momentum
(D) Same as that of latent heat

Answer 1

Hint: We will start by calculating the dimensional formula for each parameter which are pressure, work, momentum, and latent heat then we will compare each of them with the given relation $\dfrac{A}{B} = m$ , where m is linear mass density and A is force and we will obtain the dimensional formula of quantity B.

 Complete step by step answer:
Dimensional formula of any quantity can be given by fundamental quantities which can be expressed by
$ \Rightarrow mass = M$
$ \Rightarrow time = T$
$ \Rightarrow length = L$
And together we can represent them as
$ \Rightarrow [{M^a}{L^b}{T^c}]$
where a,b,c are powers of the quantities
Now from the given relation of A and B,
$ \Rightarrow \dfrac{A}{B} = m$
$ \Rightarrow B = \dfrac{A}{m}$ ---------- (1)
Where A is force and the unit of force can be given as $\dfrac{{kg \times m}}{{{s^2}}}$ hence dimensional formula can be given as
$ \Rightarrow A = [{M^1}{L^1}{T^{ - 2}}]$
And the dimensional formula of linear mass density is
 $ \Rightarrow m = [{M^1}{L^{ - 1}}]$
Substituting the values of A and m in equation (1), we get
$ \Rightarrow B = \dfrac{{[{M^1}{L^1}{T^{ - 2}}]}}{{[{M^1}{L^{ - 1}}]}}$
$\therefore B = [{M^0}{L^2}{T^{ - 2}}]$
Now we will obtain the dimensional formula for pressure, momentum, latent heat
The unit of pressure is
$ \Rightarrow kg \times {m^{ - 1}} \times {s^{ - 2}}$
So the dimensional formula of pressure is
$ \Rightarrow [{M^1}{L^{ - 1}}{T^{ - 2}}]$
The unit of momentum is
$ \Rightarrow kg \times m \times {s^{ - 1}}$
So the dimensional formula of pressure is
 $ \Rightarrow [{M^1}{L^1}{T^{ - 1}}]$
The unit of latent heat can be given by the ratio of heat energy to the mass
$ \Rightarrow L = \dfrac{Q}{M}$
Where Q is heat energy and M is mass, so the unit of latent heat will be
$ \Rightarrow \dfrac{{kg \times {m^2} \times {s^{ - 2}}}}{{kg}}$
$ \Rightarrow {m^2} \times {s^{ - 2}}$
So the dimensional formula for latent heat will be
$ \Rightarrow [{M^0}{L^2}{T^{ - 2}}]$
Now on comparing the dimensional formula of B with every given quantity force, pressure, momentum, and latent heat we obtain that the dimensional formula of latent heat and quantity B is the same.

 So the option (D) is the correct answer.

 Note: while calculating dimensional formula we have to express the units of quantities in the KGS system. The same method can be applied to find the dimensional formula of any quantity by using the units of quantities in the KGS system and representing them in terms of parameters L, M, T etc.