Question

The energy $ E $ of a system is a function of time $ t $ and is given by: $$ E(t) = \alpha t - \beta t^3 $$ where $ \alpha $ and $ \beta $ are constants. Find the dimensions of $ \alpha $ and $ \beta $.

• A. \([ML^2T^{-1}]\) and \([ML^2T]\)
• B. \([LT^{-1}]\) and \([LT]\)
• C. \([ML^2T^{-3}]\) and \([ML^2T^{-5}]\)
• D. \([MLT^{-1}]\) and \([MLT]\)

Hint:

When adding or subtracting physical quantities, each term must have the same dimension. Use this to isolate and solve for the dimensions of constants.

Answer 1

The Correct Option is C

We are given: \[ E = \alpha t - \beta t^3 \] Since both \( \alpha t \) and \( \beta t^3 \) are terms of the same expression for energy, their dimensions must be the same as that of energy:
\[ [E] = [ML^2T^{-2}] \] Let’s analyze term-by-term: 1. \( \alpha t \Rightarrow [\alpha][T] = [ML^2T^{-2}] \Rightarrow [\alpha] = \frac{[ML^2T^{-2}]}{[T]} = [ML^2T^{-3}] \)
2. \( \beta t^3 \Rightarrow [\beta][T^3] = [ML^2T^{-2}] \Rightarrow [\beta] = \frac{[ML^2T^{-2}]}{T^3} = [ML^2T^{-5}] \)