The formula \( a = -\omega x^2 \) is discussed in relation to simple harmonic motion (SHM), where the correct form should be \( a = -\omega^2 x \). To prove the formula, one must start from the equation of motion \( m a = -k x \) and relate it to the definition of angular frequency \( \omega \), which is defined as \( \omega = \sqrt{\frac{k}{m}} \). By substituting this definition into the equation of motion, one can derive the correct acceleration formula for SHM. The discussion emphasizes the importance of correctly identifying the relationship between acceleration, displacement, and angular frequency in SHM. Understanding these connections is crucial for validating the formula.
