Question

The acceleration of a point particle is given by the equation \[ \frac{d^2\mathbf{x}}{dt^2} = \alpha \frac{\mathbf{x}}{|\mathbf{x}|^7} + \beta \frac{d\mathbf{x}}{dt} \] where $\mathbf{x}$ denotes position and $t$ denotes time. Which of the following relations show the correct dimensions for $\alpha$ and $\beta$?

• A. $[\alpha] = [\text{M}^0\text{L}^7\text{T}^{-2}]$, $[\beta] = [\text{M}^0\text{L}^0\text{T}^{-1}]$
• B. $[\alpha] = [\text{M}^1\text{L}^6\text{T}^{-2}]$, $[\beta] = [\text{M}^0\text{L}^0\text{T}^{-3}]$
• C. $[\alpha] = [\text{M}^0\text{L}^6\text{T}^{-1}]$, $[\beta] = [\text{M}^0\text{L}^1\text{T}^{-2}]$
• D. $[\alpha] = [\text{M}^0\text{L}^7\text{T}^{-2}]$, $[\beta] = [\text{M}^0\text{L}^0\text{T}^0]$

Hint:

The principle of dimensional homogeneity requires that every term separated by a plus or minus sign in an equation must have the exact same dimensions as the term on the other side of the equal sign.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
By the principle of dimensional homogeneity, all terms in a sum must have the same dimensions as the result (acceleration).

Step 2: Detailed Explanation:
1. Left-Hand Side (Acceleration): Dimensions = \( [LT^{-2}] \).
2. Dimension of \( \alpha \) term:
\[ [\alpha] \frac{[L]}{[L^{7}]} = [LT^{-2}] \implies [\alpha] [L^{-6}] = [LT^{-2}] \implies [\alpha] = [L^{7}T^{-2}] \]
3. Dimension of \( \beta \) term:
\[ [\beta] [LT^{-1}] = [LT^{-2}] \implies [\beta] = [T^{-1}] \]

Step 3: Final Answer:
The dimensions are \( [\alpha] = [L^{7}T^{-2}] \) and \( [\beta] = [T^{-1}] \). This matches option (A).