91Ó°ÊÓ

Angular frequency, represented by \(\omega\), is a key concept in simple harmonic motion. It depicts how quickly the oscillation occurs. Unlike the typical frequency, which counts the number of oscillations per second, angular frequency tells us how many radians per second an object oscillating in simple harmonic motion will cover. For a tuning fork vibrating at 440 Hz, we calculate angular frequency using the relationship: \[ \omega = 2 \pi f \] Here, \(f\) is the regular frequency in hertz (Hz). Given that \(f = 440\), we can find \(\omega\): \[ \omega = 2 \pi \times 440 = 880 \pi \] So, the angular frequency for this tuning fork is \880\pi, or approximately \2764\ \text{radians per second}\.

Amplitude, denoted by \(\text{a}\) in our equation, is another crucial concept in simple harmonic motion. Amplitude is the maximum displacement from the equilibrium position during oscillation. For the given tuning fork, it's given that the maximum displacement, or amplitude, is 0.01 millimeters. This value indicates how far the end of the tuning fork moves from its central position at maximum oscillation. In mathematical terms, amplitude does not affect the frequency but defines the peak value of displacement in the function \[ d(t) = a \sin (\omega t) \] So, if \(a = 0.01\mm\), the tuning fork's motion is stretched within this range.

Frequency, symbolized by \(\text{f}\), measures how many cycles of oscillation occur per second. In the context of the tuning fork problem, the frequency is 440 Hz, indicating that the tuning fork completes 440 cycles every second. This is a crucial value because it determines the pitch of the sound produced by the fork. Frequency is related to angular frequency \(\omega \) by the equation: \[ \omega = 2 \pi f \] You can see that knowing the frequency helps directly in calculating other phenomena in harmonic motion, such as angular frequency. In our case, since \403Hz \, the angular frequency turns out to be \880 \pi\.