Qn2 Supposing that [period I] = [mx][iy][9z]. a) Find the value of: i) x, ii) y, iii) z. b) Find the relationship for the above expression on Q2.

The dimensions of the quantities are:

• Period $[T] = [T]$
• Mass $[m] = [M]$
• Length $[l] = [L]$
• Acceleration due to gravity $[g] = [LT^{-2}]$

The given relationship is $[period] = [m^x][l^y][g^z]$.
Substituting the dimensions:

$$[T] = [M^x][L^y][(LT^{-2})^z]$$ $$[M^0 L^0 T^1] = [M^x L^y L^z T^{-2z}]$$ $$[M^0 L^0 T^1] = [M^x L^{y+z} T^{-2z}]$$

Now, we equate the exponents of M, L, and T on both sides:
For M: $x = 0$
For L: $y+z = 0$
For T: $1 = -2z$

$$Period = C \cdot m^x l^y g^z$$ $$Period = C \cdot m^0 l^{1/2} g^{-1/2}$$ $$Period = C \cdot 1 \cdot \sqrt{l} \cdot \frac{1}{\sqrt{g}}$$ $$Period = C \sqrt{\frac{l}{g}}$$

The relationship is $\boxed{Period = C \sqrt{\frac{l}{g}}}$, where $C$ is a dimensionless constant.