A) What is a homogeneous equation? B) Determine the values of alpha and beta for the relation T = Ck^alpha m^beta where T is period, k is spring constant, and m is mass.

• The dimension of period $T$ is time: $[T]$.
• The constant $C$ is dimensionless: $[1]$.
• The dimension of mass $m$ is: $[M]$.
• The dimension of spring constant $k$: From Hooke's Law, $F = kx$, so $k = \frac{F}{x}$. The dimension of force $F$ is $[M][L][T]^{-2}$. The dimension of length $x$ is $[L]$. Therefore, the dimension of $k$ is $\frac{[M][L][T]^{-2}}{[L]} = [M][T]^{-2}$.

Combine the powers of the same fundamental dimensions:

For mass $[M]$:

For time $[T]$:

From equation (2):

Substitute the value of $\alpha$ into equation (1):

The values are $\alpha = -\frac{1}{2}$ and $\beta = \frac{1}{2}$.
The relation is $T = Ck^{-1/2}m^{1/2} = C\sqrt{\frac{m}{k}}$.

The values of $\alpha$ and $\beta$ are $\boxed{\alpha = -\frac{1}{2}, \beta = \frac{1}{2}}$.