Question

If \(E =\) energy, \(G =\) gravitational constant, \(I =\) impulse and \(M =\) mass, then dimensions of \( \frac{EI}{GM^2} \) are same as that of:

• A. \( \text{time} \)
• B. \( \text{mass} \)
• C. \( \text{length} \)
• D. \( \text{force} \)

Hint:

Always reduce dimensions step-by-step by cancelling powers carefully.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the dimensions of the given expression, we need to substitute the fundamental dimensional formulas for each physical quantity involved.
Step 2: Key Formula or Approach:
The dimensional formulas for the given quantities are:
1. Energy (\(E\)): \([ML^{2}T^{-2}]\)
2. Gravitational Constant (\(G\)): Derived from \(F = \frac{Gm_{1}m_{2}}{r^{2}}\) as \([M^{-1}L^{3}T^{-2}]\)
3. Impulse (\(I\)): \([MLT^{-1}]\)
4. Mass (\(M\)): \([M]\)
: Detailed Explanation:
Substitute these dimensions into the expression \(\frac{GIM^{2}}{E^{2}}\):
\[ \left[ \frac{GIM^{2}}{E^{2}} \right] = \frac{[M^{-1}L^{3}T^{-2}] [MLT^{-1}] [M]^{2}}{[ML^{2}T^{-2}]^{2}} \]
Multiply the terms in the numerator:
\[ \text{Numerator} = [M^{-1+1+2} L^{3+1} T^{-2-1}] = [M^{2}L^{4}T^{-3}] \]
Simplify the denominator:
\[ \text{Denominator} = [M^{2}L^{4}T^{-4}] \]
Divide the results:
\[ \frac{[M^{2}L^{4}T^{-3}]}{[M^{2}L^{4}T^{-4}]} = [M^{2-2}L^{4-4}T^{-3-(-4)}] = [T^{1}] \]
The dimension is that of times.
Step 3: Final Answer:
The dimensions are equivalent to time.