Question

If we choose velocity V, length L and force F as fundamental physical quantities then how would you express power in terms of V, L and F?

• A. \(F^1 L^0 V^1\)
• B. \(F^1 L^{-1} V^1\)
• C. \(F^1 L^{-1} V^2\)
• D. \(F^1 L^{-2} V^3\)

Hint:

A much faster method is to use known physical relationships. We know that Power is the dot product of Force and Velocity: \(P = \vec{F} \cdot \vec{v}\). Dimensionally, this is simply \([P] = [F][V]\). This directly gives the answer as \(F^1 V^1 L^0\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are asked to find the dimensional formula for Power (P) using Force (F), Velocity (V), and Length (L) as the fundamental units instead of Mass (M), Length (L), and Time (T).

Step 2: Key Formula or Approach:
We will use the method of dimensional analysis.
1. Write the dimensions of Power and the new fundamental quantities in terms of the standard M, L, T system.
- Power \([P] = [ML^2T^{-3}]\)
- Force \([F] = [MLT^{-2}]\)
- Velocity \([V] = [LT^{-1}]\)
- Length \([L] = [L]\)
2. Assume that Power is related to F, V, and L by the equation \(P = k F^a V^b L^c\), where k is a dimensionless constant and a, b, c are the powers we need to find.
3. Equate the dimensions on both sides and solve for a, b, and c.

Step 3: Detailed Explanation:
Set up the dimensional equation:
\[ [P] = [F]^a [V]^b [L]^c \]
Substitute the standard dimensions:
\[ [ML^2T^{-3}] = [MLT^{-2}]^a [LT^{-1}]^b [L]^c \]
\[ [M^1L^2T^{-3}] = [M^a L^a T^{-2a}] [L^b T^{-b}] [L^c] \]
Combine the powers on the right side:
\[ [M^1L^2T^{-3}] = [M^a L^{a+b+c} T^{-2a-b}] \]
Now, equate the powers of M, L, and T from both sides:
- For M: \(a = 1\)
- For T: \(-2a - b = -3\)
- For L: \(a + b + c = 2\)
Solve the system of equations:
From the M equation, we have \(a = 1\).
Substitute \(a=1\) into the T equation:
\[ -2(1) - b = -3 \implies -2 - b = -3 \implies b = 1 \]
Substitute \(a=1\) and \(b=1\) into the L equation:
\[ 1 + 1 + c = 2 \implies 2 + c = 2 \implies c = 0 \]
So, the powers are \(a=1, b=1, c=0\). The expression for Power is \(F^1 V^1 L^0\).

Step 4: Final Answer:
Power can be expressed as \(F^1 L^0 V^1\).