91Ó°ÊÓ

In the context of simple harmonic motion (SHM), angular frequency (\(\omega\)) plays a pivotal role. It tells us how fast the object is moving through its cycle of motion. Specifically, it is the rate at which the object oscillates in radians per second.
\[\omega = \frac{v_{max}}{a_{max}/v_{max}}\]
Here, \(v_{max}\) is the maximum speed, and \(a_{max}\) is the maximum acceleration. For our exercise, we compute \(\omega\) with the given values for \(v_{max}\) and \(a_{max}\), resulting in:

• \(v_{max} = 1.25 \, \text{m/s}\)
• \(a_{max} = 6.89 \, \text{m/s}^2\)

This gives us \(\omega = \frac{1.25}{6.89/1.25}\). Understanding \(\omega\) helps us determine how quickly the cycle completes and is crucial for calculating the period of the oscillation.

The period of oscillation, denoted as \(T\), is the time it takes for an object to complete one full cycle of its motion. It's directly linked to the angular frequency by the formula:

• \(T = \frac{2 \pi}{\omega}\)

This period is essentially a measure of time. With \(\omega\) being calculated from the exercise, we find \(T\) using the formula above. The relationship indicates that as the angular frequency increases, the period decreases, meaning the cycle is completed quicker.
Finding the period allows us to determine key time intervals in the motion, such as how quickly the SHM object transitions from maximum speed to maximum acceleration. For our task, we specifically look at \(\frac{1}{4}T\) to get the time elapsing between these states.

The amplitude \(A\) in simple harmonic motion is the maximum displacement from the equilibrium position. It determines the extent of the object's movement in the cycle. It's crucial because it's directly linked to both maximum speed \(v_{max}\) and maximum acceleration \(a_{max}\):

• \(v_{max} = \omega A\)
• \(a_{max} = \omega^2 A\)

From these relationships, we can see that the amplitude affects the maximum potential values the object can reach in speed and acceleration.
While \(A\) isn't explicitly solved in every problem, knowing how to determine it from the given maximum speed or acceleration can be crucial in understanding the motion's dynamics and ensuring the calculations are consistent when solving for \(\omega\).

Maximum speed \(v_{max}\) and maximum acceleration \(a_{max}\) are key parameters in SHM. The maximum speed is observed at the equilibrium position, where the object moves unhindered by forces changing its motion direction. It is calculated using the formula:

• \(v_{max} = \omega A\)

Maximum acceleration, on the other hand, occurs at the extremes of motion, where the object is furthest from equilibrium. It is calculated using:

• \(a_{max} = \omega^2 A\)

These formulas tie the dynamics of the motion together by relating them to \(\omega\) and \(A\).
Understanding these parameters helps us predict how the oscillating system behaves. In this scenario, analyzing maximum speed and acceleration allows us to calculate \(\omega\) and eventually determine time intervals like \(\Delta t = \frac{T}{4}\), showing how quickly these maximal states change during motion.