if power P and linear mass density are related as `P=alpha/(beta^2+lambda^2)`, then find the dimensions of `alpha` and `beta`

Answer

To solve the problem, we need to find the dimensions of the quantities \( \alpha \) and \( \beta \) given the relationship \( P = \frac{\alpha}{\beta^2 + \lambda^2} \), where \( P \) is power and \( \lambda \) is linear mass density.

### Step-by-Step Solution:

1. **Identify the dimensions of power \( P \)**:
- Power is defined as the rate of doing work or energy transfer per unit time. The dimension of energy is \( [E] = [M L^2 T^{-2}] \).
- Therefore, the dimension of power \( P \) is:
\[ [P] = \frac{[E]}{[T]} = \frac{[M L^2 T^{-2}]}{[T]} = [M L^2 T^{-3}] \]

2. **Identify the dimensions of linear mass density \( \lambda \)**:
- Linear mass density \( \lambda \) is defined as mass per unit length. Thus, its dimension is:
\[ [\lambda] = \frac{[M]}{[L]} = [M L^{-1}] \]

3. **Relate the dimensions of \( \beta \) to \( \lambda \)**:
- Since \( \beta^2 \) and \( \lambda^2 \) are added together in the expression \( P = \frac{\alpha}{\beta^2 + \lambda^2} \), they must have the same dimensions.
- Therefore, the dimension of \( \beta^2 \) must equal the dimension of \( \lambda^2 \):
\[ [\beta^2] = [\lambda^2] = [M^2 L^{-2}] \]
- Taking the square root gives us the dimension of \( \beta \):
\[ [\beta] = [M^{1} L^{-1}] = [M L^{-1}] \]

4. **Substituting dimensions into the equation**:
- Now we substitute the dimensions into the equation \( P = \frac{\alpha}{\beta^2 + \lambda^2} \).
- Since \( \beta^2 \) has the dimension \( [M^2 L^{-2}] \), we can rewrite the equation as:
\[ [P] = \frac{[\alpha]}{[M^2 L^{-2}]} \]

5. **Rearranging to find the dimension of \( \alpha \)**:
- Rearranging gives:
\[ [\alpha] = [P] \cdot [M^2 L^{-2}] = [M L^2 T^{-3}] \cdot [M^2 L^{-2}] = [M^3 L^0 T^{-3}] \]
- Therefore, the dimension of \( \alpha \) is:
\[ [\alpha] = [M^3 T^{-3}] \]

### Final Results:
- The dimensions of \( \alpha \) and \( \beta \) are:
- \( [\alpha] = [M^3 T^{-3}] \)
- \( [\beta] = [M L^{-1}] \)