91Ó°ÊÓ

Angular frequency, represented by \( \omega \), is a fundamental characteristic in Simple Harmonic Motion (SHM). It describes how quickly the object oscillates back and forth.
The formula to calculate angular frequency is given by \( \omega = \frac{2\pi}{T} \), where \( T \) is the period, the time it takes to complete one full cycle of the motion. For our bolt, the period \( T \) is 1.500 seconds. Plugging this into our formula, we calculate the angular frequency as \( \omega \approx 4.1888 \, \text{rad/s} \).
This value of \( \omega \) signifies that the bolt completes the equivalent of \( 4.1888 \) radians per second during its oscillation. Angular frequency is essential because it ties together time, displacement, and force in SHM, providing a complete picture of the motion's characteristics.

In SHM, the displacement of an object from its mean position at any time is given by the equation \( x(t) = A \cos(\omega t) \), where \( A \) is the amplitude, or the maximum displacement from the mean. Here, \( A = 0.240 \, \text{m} \).
To find the displacement of the bolt at a specific time, say \( t = 0.500 \, \text{s} \), we substitute \( t \) and the angular frequency \( \omega \) into our displacement equation. This gives \( x(0.500) = 0.240 \cos(4.1888 \times 0.500) \approx -0.120 \, \text{m} \).
The negative displacement indicates the bolt is on the opposite side of its starting position, moving towards the equilibrium point. Understanding displacement helps visualize how far and in which direction the object has moved at any given time during its oscillation.

In SHM, the force acting on an object is always directed towards the equilibrium position and is proportional to the displacement from it. This force can be found using Hooke's Law for SHM, expressed mathematically as \( F = -m\omega^2 x \).
For our bolt, with mass \( m = 0.0200 \, \text{kg} \), angular frequency \( \omega = 4.1888 \, \text{rad/s} \), and displacement \( x = -0.120 \, \text{m} \), we calculate the force as \( F \approx 0.0421 \, \text{N} \).
The positive sign in the result indicates that the force is in the positive direction, meaning it is acting towards the original position of equilibrium. Grasping how force is calculated in SHM aids in understanding the restoring nature of SHM, where the force aims to bring the object back to the center of motion.

The velocity of an object in SHM can tell us a lot about its energy state and speed at different positions during the cycle. It is given by the equation \( v = \omega \sqrt{A^2 - x^2} \).
For the bolt, with amplitude \( A = 0.240 \, \text{m} \) and displacement \( x = -0.180 \, \text{m} \), the velocity at this position can be calculated. Using the already found angular frequency \( \omega = 4.1888 \, \text{rad/s} \), the velocity becomes \( v \approx 0.642 \, \text{m/s} \).
This indicates how fast the bolt is moving at that point and helps us understand the dynamic motion at various stages of the SHM process. By determining the velocity, one can analyze not just the speed, but also gain insights into the kinetic energy changes as the object oscillates.