The needle of a sewing machine moves in SHM with an amplitude of 1.2 cm and a frequency of 8 Hz…

Step 1. $\omega = 2\pi \times 8\ \text{Hz} = 50.3\ \text{rad/s}$.

Step 2. Speed. $|v| = 50.3\ \text{rad/s} \times \sqrt{(0.012\ \text{m})^2 - (0.006\ \text{m})^2} = 50.3\ \text{rad/s} \times 0.0104\ \text{m} = 0.522\ \text{m/s}$.

Step 3. Acceleration. $|a| = \omega^2 x = (50.3\ \text{rad/s})^2 \times 0.006\ \text{m} = 15.2\ \text{m/s}^2$, towards the centre.