Question

If velocity [V] time [T] and force [F] are chosen as the base quantities, the dimensions of the mass will be :

• A. $[FT^{-1}V^{-1}]$
• B. $[FVT^{-1}]$
• C. $[FT^2V]$
• D. $[FTV^{-1}]$

Hint:

To quickly find dimensions of a quantity in terms of a new set of base units, always look for the simplest fundamental equation connecting all the involved variables. Here, \(F = m \cdot \frac{v}{t}\) is the most direct link.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In the study of dimensions, we can express any derived physical quantity in terms of chosen base quantities by establishing a relationship between them using Newton's second law of motion or basic definitions.
Step 2: Key Formula or Approach:
According to Newton's Second Law, Force (\(F\)) is defined as the rate of change of momentum, or more simply, the product of mass (\(m\)) and acceleration (\(a\)):
\[ F = m \times a \]
Since acceleration (\(a\)) is the rate of change of velocity (\(V\)) with respect to time (\(T\)), we can write:
\[ a = \frac{V}{T} \]
Step 3: Detailed Explanation:
Substitute the expression for acceleration into the force equation:
\[ F = m \times \left(\frac{V}{T}\right) \]
To find the dimensions of mass (\(m\)) in terms of \(F\), \(V\), and \(T\), we rearrange the formula to isolate \(m\):
\[ m = \frac{F \times T}{V} \]
Now, expressing this in dimensional notation:
\[ [m] = [F] [T] [V]^{-1} \]
\[ [m] = [FTV^{-1}] \]
Step 4: Final Answer:
Comparing this result with the given options, we find that the dimensions of mass are \([FTV^{-1}]\).