91Ó°ÊÓ

Understanding angular frequency is crucial in simple harmonic motion. It's denoted by the symbol \( \omega \) and represents how fast an object oscillates within a given cycle. The useful formula for calculating angular frequency is: \[ \omega = 2\pi f \] where \( f \) is the frequency in hertz (Hz). It's essential to know that this formula will convert linear frequency into angular frequency, expressed in radians per second (rad/s). In our exercise, the diaphragm is moving with a frequency of \( 440 \ Hz \), and when we apply the formula, we end up with: \[ \omega = 880\pi \ \text{rad/s} \] This value tells us how rapidly the diaphragm completes its oscillation cycle in terms of angular displacement.

When an object like our diaphragm oscillates, it reaches a maximum speed at the point of equilibrium. This maximum speed is derived using the equation: \[ v_{max} = \omega A \] where \( \omega \) is the angular frequency, and \( A \) is the amplitude or maximum displacement. For our case, the amplitude is \( 0.00075 \ m \). With the calculated \( \omega \) value, substituting gives us: \[ v_{max} = 2.07 \ \text{m/s} \] This formula yields the peak speed as the diaphragm passes through its midpoint. Remember, in harmonic motion, speed is highest at the equilibrium point since kinetic energy is fully transformed.

In simple harmonic motion, maximum acceleration happens at the points of maximum displacement, explained by the formula: \[ a_{max} = \omega^2 A \] The acceleration relates to both the square of the angular frequency and the amplitude, illustrating how quickly the speed changes at the extremes of the oscillation. Applying this to our diaphragm with \( \omega = 880\pi \ \text{rad/s} \) and \( A = 0.00075 \ m \), we find: \[ a_{max} = 5820 \ \text{m/s}^2 \] This expresses the strongest rate of change in speed when the diaphragm reaches its outermost positions. Comprehending this helps us predict how forces might act on oscillating objects.