91Ó°ÊÓ

In simple harmonic motion, one important concept is the angular frequency, denoted by \(\textbf{\omega}\). This tells us how many radians the object moves through per second. Angular frequency is related to the period \(\textbf{T}\) of the oscillation by the formula: \[ \omega = \frac{2\pi}{T} \] The period is the time it takes for one complete cycle of motion. So, using the given period \(0.500 \text{ s}\), we substitute it into the formula to find \(\textbf{\omega}\): \[ \omega = \frac{2\pi}{0.500} = 4\pi \text{ rad/s} \] This means that every second, the watch's balance wheel moves through \(4\pi\) radians.

The rotational speed of an oscillating object shows how quickly it moves at any given point in time. The maximum rotational speed, \( \textbf{\omega_{max}} \), occurs at the equilibrium position (where the displacement is zero). It can be calculated using the formula: \[ \omega_{max} = \omega A \] Here, \(A\) is the amplitude of motion. Given \(A = \pi\text{ rad}\) and \( \omega = 4\pi \text{ rad/s}\), we have: \[ \omega_{max} = (4\pi)(\pi) = 4\pi^2 \text{ rad/s} \] This is the fastest the wheel moves. To find the rotational speed when the displacement is \(\pi / 2 \text{ rad}\), we use the energy conservation relation: \[ \omega_x = \omega \sqrt{A^2 - x^2} \] With \( x = \pi / 2 \text{ rad} \) and \( A = \pi \text{ rad} \), we get: \[ \omega_x = 4\pi \sqrt{\pi^2 - (\pi / 2)^2} = 4\pi \sqrt{\pi^2 - \pi^2 / 4} = 4\pi \sqrt{3\pi^2 / 4} = 4\pi (\frac{\pi\sqrt{3}}{2}) = 2\pi^2\sqrt{3} \text{ rad/s} \] This formula provides the rotational speed at different points of the oscillation.

Rotational acceleration, represented by \(\textbf{\alpha}\), shows how quickly the rotational speed changes. It can be determined using the formula: \[ \alpha = -\omega^2 x \] Negative sign indicates that the acceleration is directed opposite to the displacement. Using \( \omega = 4\pi \text{ rad/s} \) and displacement \( x = \pi / 4 \text{ rad} \): \[ \alpha = -(4\pi)^2 (\pi / 4) = -16\pi^2(\pi / 4) = -4\pi^3 \text{ rad/s}^2 \] This formula gives the magnitude of rotational acceleration at a specific point in simple harmonic motion.