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 \(F\)?
Show Hint
Power can be written as \(P=FV\), because power is force multiplied by velocity.
Step 1: Understanding the Concept:
This problem requires using dimensional analysis to express a derived quantity (Power) in terms of a new set of fundamental quantities (Force, Length, Velocity). We assume Power is proportional to some powers of F, L, and V, and then equate the dimensions on both sides to find the exponents. Step 2: Key Formula or Approach:
1. Write down the dimensions of all quantities involved in terms of the standard fundamental dimensions M (Mass), L (Length), and T (Time).
- Power [P] = $[ML^2T^{-3}]$
- Force [F] = $[MLT^{-2}]$
- Length [L] = $[L]$
- Velocity [V] = $[LT^{-1}]$
2. Assume Power is related to F, L, and V by the equation: $[P] = [F]^a [L]^b [V]^c$.
3. Substitute the dimensions and solve for the exponents a, b, and c. Step 3: Detailed Explanation:
Let's assume Power, P, is expressed as:
\[ P = k F^a L^b V^c \]
where k is a dimensionless constant. Equating the dimensions on both sides:
\[ [P] = [F]^a [L]^b [V]^c \]
Substitute the standard dimensions:
\[ [ML^2T^{-3}] = [MLT^{-2}]^a [L]^b [LT^{-1}]^c \]
\[ [M^1 L^2 T^{-3}] = [M^a L^a T^{-2a}] [L^b] [L^c T^{-c}] \]
Combine the powers of M, L, and T on the right side:
\[ [M^1 L^2 T^{-3}] = [M^a L^{a+b+c} T^{-2a-c}] \]
Now, equate the exponents for each dimension:
1. For M: $1 = a$
2. For T: $-3 = -2a - c$
3. For L: $2 = a+b+c$
From (1), we have $a=1$.
Substitute $a=1$ into (2):
\[ -3 = -2(1) - c \implies -3 = -2 - c \implies c = 1 \]
Substitute $a=1$ and $c=1$ into (3):
\[ 2 = 1 + b + 1 \implies 2 = 2 + b \implies b = 0 \]
So, the exponents are $a=1, b=0, c=1$.
The expression for power is $P = k F^1 L^0 V^1$.
Alternatively, we can use the definition of power: Power = Force $\times$ Velocity.
\[ P = F \cdot V \]
This directly gives the relationship $P = F^1 V^1$, which can be written as $F^1 L^0 V^1$. Step 4: Final Answer:
Power can be expressed as $F^1 L^0 V^1$. Therefore, option (A) is correct.
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