Question

If force (F), length (L) and time (T) are taken as the fundamental quantities. Then what will be the dimension of density:

• A. [FL\(^{-3}\)T\(^{3}\)]
• B. [FL\(^{-5}\)T\(^{2}\)]
• C. [FL\(^{-4}\)T\(^{2}\)]
• D. [FL\(^{-3}\)T\(^{2}\)]

Hint:

This type of dimensional analysis problem is common. The key is always to find an expression for the old fundamental quantities (like M) in terms of the new ones (F, L, T) using defining equations (like F=ma). Then substitute back into the target quantity's formula.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are asked to express the dimensions of density in a new system where Force (F), Length (L), and Time (T) are fundamental, instead of the usual Mass (M), Length (L), and Time (T).
Step 2: Key Formula or Approach:
1. Write the dimensional formula of density in the standard [MLT] system.
2. Write the dimensional formula of the new fundamental quantity (Force) in the [MLT] system.
3. Use the relation from step 2 to express Mass [M] in terms of [F], [L], and [T].
4. Substitute this expression for [M] into the dimensional formula for density.
Step 3: Detailed Explanation:
1. Dimension of Density (\(\rho\)):
Density is defined as mass per unit volume.
\[ [\rho] = \frac{[\text{Mass}]}{[\text{Volume}]} = \frac{[\text{M}]}{[\text{L}^3]} = [\text{ML}^{-3}] \] 2. Dimension of Force (F):
From Newton's second law, Force = Mass \(\times\) Acceleration.
\[ [\text{F}] = [\text{M}] \times [\text{LT}^{-2}] = [\text{MLT}^{-2}] \] 3. Express [M] in terms of [F], [L], [T]:
From the dimension of force, we can rearrange to solve for [M].
\[ [\text{M}] = \frac{[\text{F}]}{[\text{LT}^{-2}]} = [\text{FL}^{-1}\text{T}^{2}] \] 4. Substitute for [M] in the dimension of density:
Now, we substitute the new expression for [M] into the dimensional formula for density.
\[ [\rho] = [\text{M}][\text{L}^{-3}] = ([\text{FL}^{-1}\text{T}^{2}])([\text{L}^{-3}]) \] Combining the powers of L:
\[ [\rho] = [\text{FL}^{-1-3}\text{T}^{2}] = [\text{FL}^{-4}\text{T}^{2}] \] Step 4: Final Answer:
The dimension of density in the [FLT] system is [FL\(^{-4}\)T\(^{2}\)].