91Ó°ÊÓ

Angular frequency, denoted by the Greek letter \( \omega \), plays a crucial role in describing simple harmonic motion (SHM). It tells us how quickly the object oscillates in radians per second. You can think of it as the 'speed' of the oscillation.
For a system with a known period \( T \), the angular frequency is calculated using the formula:

• \( \omega = \frac{2\pi}{T} \)

Here, \( 2\pi \) represents a full oscillation or cycle in radians, and \( T \) is the duration of one complete cycle.
For instance, in the problem where \( T = 0.73 \) seconds, we plug this into the formula to get \( \omega \approx 8.61 \) s\( ^{-1} \). This means that the mass makes about 8.61 cycles per second in terms of angular displacement.

The displacement equation in simple harmonic motion helps us track the position of an oscillating object over time. It's written as:

• \( x(t) = A \cos(\omega t + \phi) \)

In this equation, \( A \) is the amplitude, the maximum displacement from the equilibrium position. The amplitude tells us how far the object moves from the center.
For our problem, \( A = 5.4 \) cm signifies that the object moves 5.4 cm from its rest position at its peak.
The symbol \( \omega \) is the angular frequency, **previously calculated as 8.61 s\( ^{-1} \)**, determining how fast the oscillation occurs.\( t \) is time, a variable that shows how the displacement changes as time progresses. \( \phi \) is the phase angle, which we'll dive into shortly.
By substituting our values into the equation, we write it as \( x(t) = 5.4\cos(8.61t) \), providing us a precise description of the motion across time.

The phase angle, \( \phi \), is an important parameter that determines the starting point of the oscillation in the displacement equation. By setting \( \phi \), you establish the initial position of the oscillating object at time \( t = 0 \).
In simple harmonic motion, different initial conditions result in different phase angles. If an object starts from its maximum amplitude, like in our example, the phase angle should be 0.
This is because cosine of zero is one, indicating the object is at its peak displacement when time starts. The phase angle adjusts how the cosine wave aligns with time zero. When \( \phi = 0 \), the equation simplifies to \( x(t) = A \cos(\omega t) \), translating directly to the start of motion at the maximum displacement.
Thus, in our exercise, \( \phi = 0 \) ensures the displacement equation, \( x(t) = 5.4\cos(8.61t) \), accurately reflects the initial conditions specified in the problem.