Given : force `= (alpha)/("density" + beta^3).` What are the dimensions of `alpha, beta ?`

Options

A. `M L^(-2) T^(-2), ML^(-1//3)`
B. `M^2 L^4 T^(-2), M^(1//3) L^(-1)`
C. `M^2 L^(-2) T^(-2), M^(1//3) L^(-1)`
D. `M^2 L^(-2), M L^(-3)`

Explanation

To find the dimensions of \( \alpha \) and \( \beta \) in the equation \( \text{force} = \frac{\alpha}{\text{density} + \beta^3} \), we will follow these steps:

### Step 1: Understand the dimensions of force and density
- The dimension of force \( F \) is given by:
\[ [F] = [M][L][T]^{-2} = MLT^{-2} \]
- The dimension of density \( \rho \) is given by:
\[ [\rho] = \frac{[M]}{[L]^3} = ML^{-3} \]

### Step 2: Analyze the equation
The equation can be rewritten as:
\[ F = \frac{\alpha}{\rho + \beta^3} \]
For the equation to be dimensionally consistent, the dimensions of \( \alpha \) must match the dimensions of \( F \) when multiplied by the dimensions of \( \rho + \beta^3 \).

### Step 3: Determine the dimensions of \( \beta \)
Since \( \beta^3 \) must have the same dimensions as density \( \rho \):
\[ [\beta^3] = [\rho] = ML^{-3} \]
Thus, we can write:
\[ [\beta]^3 = ML^{-3} \]
Taking the cube root gives us:
\[ [\beta] = M^{1/3}L^{-1} \]

### Step 4: Substitute \( \beta \) back into the equation
Now we can substitute \( \beta \) back into the equation to find \( \alpha \). Since \( \rho + \beta^3 \) has the same dimensions as \( \rho \) (as \( \beta^3 \) is of the same dimension):
\[ [\rho + \beta^3] = ML^{-3} \]

### Step 5: Rearranging the equation for \( \alpha \)
From the equation \( F = \frac{\alpha}{\rho + \beta^3} \), we can rearrange it to find \( \alpha \):
\[ \alpha = F \cdot (\rho + \beta^3) \]
Substituting the dimensions we have:
\[ [\alpha] = [F] \cdot [\rho] = (MLT^{-2}) \cdot (ML^{-3}) = M^2L^{-2}T^{-2} \]

### Final Dimensions
- The dimensions of \( \alpha \) are \( M^2L^{-2}T^{-2} \).
- The dimensions of \( \beta \) are \( M^{1/3}L^{-1} \).

### Summary
- Dimensions of \( \alpha \): \( M^2L^{-2}T^{-2} \)
- Dimensions of \( \beta \): \( M^{1/3}L^{-1} \)