A body in SHM moves at 0.96 m/s when it is 6 cm from the mean position, and at 0.72 m/s when it is…

Step 1. Two equations. In metres and m/s:

$(0.96\ \text{m/s})^2 = \omega^2\left(A^2 - (0.06\ \text{m})^2\right)$

$(0.72\ \text{m/s})^2 = \omega^2\left(A^2 - (0.08\ \text{m})^2\right)$

Step 2. Subtract. $0.9216\ \text{m}^2/\text{s}^2 - 0.5184\ \text{m}^2/\text{s}^2 = \omega^2\left(0.0064\ \text{m}^2 - 0.0036\ \text{m}^2\right)$. So $\omega^2 = \dfrac{0.4032\ \text{m}^2/\text{s}^2}{0.0028\ \text{m}^2} = 144\ \text{s}^{-2}$ and $\omega = 12\ \text{rad/s}$.

Step 3. Amplitude. From the first equation, $A^2 = (0.06\ \text{m})^2 + \left(\dfrac{0.96\ \text{m/s}}{12\ \text{rad/s}}\right)^2 = 0.0036\ \text{m}^2 + 0.0064\ \text{m}^2 = 0.01\ \text{m}^2$. So $A = 0.10\ \text{m} = 10\ \text{cm}$.

Step 4. Period. $T = \dfrac{2\pi}{12\ \text{rad/s}} = 0.524\ \text{s}$.