Question

The dimensions $\left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$ may be of
A. Pressure
B. Energy per unit volume
C. Force
D. Linear momentum

Answer 1

Hint: We are given the dimensional formula of a particular quantity. We know that any physical quantity can be expressed in terms of the seven physical quantities. We have to check that which quantity out of the four given quantities is expressed using the dimensional formula. Use the expression for the quantities and convert them into the dimensional formula. More than one quantity can have the same dimensional formula.

Complete step by step answer:
We know that pressure is the amount of force acting on a unit area.
We can write the expression for pressure as,
$P = \dfrac{F}{A}$
Where $P$ stands for the pressure, $F$stands for the force acting on a surface, and $A$stands for the area of the surface.
We know that force is the product of acceleration and mass. Hence we can write the dimensional formula for force as,
$F = \left[ {ML{T^{ - 2}}} \right]$
The dimensional formula for the area can be written as,
$A = \left[ {{L^2}} \right]$
Substituting the dimensional formulae for force and area in the expression of pressure, we get
$P = \dfrac{F}{A} = \dfrac{{\left[ {ML{T^{ - 2}}} \right]}}{{\left[ {{L^2}} \right]}} = \left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$
Since we have already calculated the dimensional formula for force while calculating for pressure, we can exclude force from the options.
Now let us calculate energy per unit volume.
We know that energy is the same as the work is done, hence we can consider energy as the product of force and displacement.
We can write energy as,
$E = Fx$
Where $E$stands for the energy, $F$ stands for the force, and $x$ stands for the displacement.
The dimensional formula for force can be written as,
$F = \left[ {ML{T^{ - 2}}} \right]$
The dimensional formula for displacement is,
$x = \left[ L \right]$
So, we can write the dimensional formula for energy as,
$E = Fx = \left[ {ML{T^{ - 2}}} \right]\left[ L \right] = \left[ {M{L^2}{T^{ - 2}}} \right]$
The energy per unit volume can be written as,
$\dfrac{E}{V} = \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {{L^3}} \right]}} = \left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$
Linear momentum is the product of mass and velocity,
The linear momentum can be written as,
$P = mv$
The dimensional formula for momentum can be written as,
$P = mv = \left[ M \right]\left[ {L{T^{ - 1}}} \right] = \left[ {ML{T^{ - 1}}} \right]$.

Option (A) and option (B): Pressure and energy per unit volume.

Note:
An equation connecting the physical quantity with dimensional formula is called the dimensional equation of that physical quantity. The dimensions of the same fundamental quantity must be the same on either side of a dimensional formula. Dimensional analysis is the analysis of an equation by expressing physical quantities in terms of their base quantities by assigning the appropriate dimensions.