91Ó°ÊÓ

Frequency in simple harmonic motion (SHM) refers to how often a system completes a full oscillation or cycle per second. In this problem, both objects A and B have different frequencies, which influence their oscillatory behavior.
For object A, the frequency is derived from its equation, \( x_A = (2.0 \text{ m}) \sin(2.0t) \). The 2.0 in the sine function indicates that object A completes 2 full cycles every \(2\pi\) seconds. In terms of frequency, it completes \(1/\pi\) cycles per second.
Similarly, object B, with equation \( x_B = (5.0 \text{ m}) \sin(3.0t) \), has a frequency of \(3/2\pi\) cycles per second. The frequency is crucial as it dictates how often each object returns to the origin (zero position).
Understanding frequency helps in predicting the motion's timing, making it essential to find when both objects reach the origin simultaneously. It's like rhythm in music; knowing it allows you to predict beats, which, in this case, are times the objects return to the start.

The sine function is central to describing simple harmonic motion, often used for its periodic properties. In the given equations, the sine function models the oscillations of objects A and B over time.
For object A, \( x_A = (2.0 \text{ m}) \sin(2.0t) \), and for object B, \( x_B = (5.0 \text{ m}) \sin(3.0t) \), the sine function dictates how each object moves back and forth. The output of the sine function fluctuates between -1 and 1, capturing the typical wave motion.
In simple harmonic motion, this wave is repeated in cycles, with the coefficients (2.0 and 3.0) scaling the angle and therefore adjusting the period and frequency. This repetition is why the sine function is an excellent choice for modeling such periodic motions. The point where the function equals zero corresponds to the objects crossing the origin. It's these properties that make the sine function so ideal for representing SHM.

Position equations express the location of objects undergoing simple harmonic motion as functions of time. In this context, they determine when objects are at any specific position in their cycle, especially when they pass through the origin.
We have two position equations in this problem: \( x_A = (2.0 \text{ m}) \sin(2.0t) \) for object A and \( x_B = (5.0 \text{ m}) \sin(3.0t) \) for object B. These equations tell us the position along their respective paths at any given time \( t \).
To find when both objects pass through the origin (position zero), we solve these equations for when their sine components equal zero. This results in times derived from the equality condition \( \sin(2.0t) = 0 \) and \( \sin(3.0t) = 0 \).

• The solution shows these times occurring every multiple of \( \pi \), leading us to figure out simultaneous zero crossings with common multiples in \( t = \frac{k\pi}{6.0} \).
• Using position equations allows us to precisely predict when and where the objects will be, highlighting the synchronization of cyclic movements in SHM.