Question

The product of energy and time is called action. The dimensional formula for action is the same as that for
A) force x velocity
B) impulse x distance
C) power
D) angular energy

Answer 1

Hint: Energy is a quantitative property that must be transferred to an object in order for it to perform work. We will find the dimensional formula of energy using the dimensional formula of kinetic energy and then multiply it with time and compare the answer with the dimensional formulae of all the choices given in the question.

Complete step by step solution:
We know that the product of energy and time is called action. Let us start by finding the dimensional formula for action. To do that we should first find the dimensional formula for energy. We can use any formula that can help us in calculating the dimensional formula for energy. Since we know the kinetic energy of an object can be calculated as $ E = \dfrac{1}{2}m{v^2} $ where $ m $ is the mass of the object and $ v $ is its velocity, we can write the dimensional formula for energy as
 $ E = {M^1} \times {\left( {{L^1}{T^{ - 1}}} \right)^2} $
 $ E = {M^1}{L^2}{T^{ - 2}} $
Now we can calculate the dimensional formula for action as
 $ {\text{Action}}\,{\text{ = }}{M^1}{L^2}{T^{ - 2}} \times {T^1} $
 $ \Rightarrow {\text{Action}}\, = {M^1}{L^2}{T^{ - 1}} $
Let’s now consider the dimensional formulae of the units given in the question:
Force x velocity $ = {M^1}{L^1}{T^{ - 2}} \times {L^1}{T^{ - 1}} $
  $ = {M^1}{L^2}{T^{ - 3}} $ which does not match the dimensions of action
Impulse x distance $ = {M^1}{L^1}{T^{ - 1}} \times {L^1} $
  $ = {M^1}{L^2}{T^{ - 1}} $ which matches the dimensions of action
C) Power $ = \dfrac{{{\text{Energy}}}}{{{\text{Time}}}} $
Since we know the dimensions of energy are $ E = {M^1}{L^2}{T^{ - 2}} $ , we can write
 $ = \dfrac{{{M^1}{L^2}{T^{ - 2}}}}{{{T^1}}} $
 $ = {M^1}{L^2}{T^{ - 3}} $
which again does not match the dimensions of action
D) Angular energy has the same dimensions as energy $ = {M^1}{L^2}{T^{ - 2}} $ which are not equal to the dimensional formula for action
The dimensional formula for Impulse x distance matches the dimensional formula for action so the correct choice is option (B).

Note:
The dimensional formula for energy can be alternatively calculated using potential energy as
 $ E = mgh $ where $ m $ is the mass of the object, $ g $ is the gravitational acceleration, and $ h $ is its height. So the dimensional formula of energy can be calculated as:
 $ E = {M^1} \times {L^1}{T^{ - 2}} \times {L^1} $
 $ \Rightarrow E = {M^1}{L^2}{T^{ - 2}} $ .