91Ó°ÊÓ

When we talk about angular frequency in simple harmonic motion, we are referring to how fast something is oscillating around a central point. The angular frequency, denoted by \( \omega \), is directly linked to the regular frequency (\( f \)), which tells us the number of oscillations per second. To find \( \omega \), we use the formula: \( \omega = 2\pi f \). Here, \( 2\pi \) comes from the fact that a full oscillation is equivalent to a full circle, which is \( 2\pi \) radians.

In this exercise, the diaphragm's frequency is given as \( 440 \mathrm{~Hz} \). Applying the formula, we calculate the angular frequency to be approximately \( 2764.6 \text{ rad/s} \). This means the diaphragm completes about \( 2764.6 \) radians in one second.

• Why radians? In mathematics and physics, radians are a natural way to measure angles. This makes them more suitable for expressions involving circular or oscillatory motion.

The maximum speed in simple harmonic motion (SHM) occurs when the object passes through the equilibrium position. At this point, all the energy is kinetic, and the speed reaches its peak. The formula for maximum speed \( v_{max} \) is: \( v_{max} = \omega A \), where \( \omega \) is the angular frequency, and \( A \) is the amplitude—the furthest point from the center.

Before plugging numbers into the formula, ensure that the amplitude is in meters for consistency with SI units. In this problem, the amplitude is \( 0.75 \text{ mm} \), which converts to \( 0.75 \times 10^{-3} \text{ m} \). Now, insert the values: \( v_{max} = 2764.6 \times 0.75 \times 10^{-3} \approx 2.073 \text{ m/s} \).

• Why is speed maximum at equilibrium? In SHM, potential energy is highest at the extremes. Thus, as the object moves towards the center, potential energy transforms into kinetic, maximizing speed at the midpoint.

Maximum acceleration is another critical aspect of SHM, occurring at the endpoints of oscillation rather than the center. This is where the speed is zero, but the force—and consequently, acceleration—is at its peak. Maximum acceleration \( a_{max} \) is calculated using the formula \( a_{max} = \omega^2 A \).

From previous calculations, we have \( \omega \approx 2764.6 \text{ rad/s} \) and \( A = 0.75 \times 10^{-3} \text{ m} \). Substitute into the formula to get: \( a_{max} = (2764.6)^2 \times 0.75 \times 10^{-3} \approx 5733.45 \text{ m/s}^2 \).

• Why is acceleration maximum at the endpoints? At the extremes, the restoring force is highest, resulting in maximum acceleration. It's like pulling on a spring, where the force increases the more you stretch it.