Problem 206 - Calculus Volume 1

Question

The displacement from rest of a mass attached to a spring satisfies the simple harmonic motion equation \(x(t)=A \cos (\omega t-\phi),\) where \(\phi\) is a phase constant, \(\omega\) is the angular frequency, and \(A\) is the amplitude. Find the average velocity, the average speed (magnitude of velocity), the average displacement, and the average distance from rest (magnitude of displacement) of the mass.

Step-by-Step Solution

Answer

Average velocity: 0; Average speed: \(\frac{2A\omega}{\pi}\); Average displacement: 0; Average distance from rest: \(\frac{2A}{\pi}\).

Step 1: Understand the Average Velocity

Average velocity is defined as the total displacement divided by the total time. Since the displacement is periodic with a period of one complete oscillation, the mass will return to its original position after one period. Hence, the net displacement over one full oscillation is zero. Thus, the average velocity over one complete period is zero.

Step 2: Understand the Average Speed

Average speed is defined as the total distance traveled divided by the total time. For one complete cycle, the mass travels a distance of \(4A\), which is four times the amplitude (twice moving outwards and twice returning back). The time for this complete cycle is the period \(T = \frac{2\pi}{\omega}\). Therefore, the average speed is \(\frac{4A}{T} = \frac{4A\omega}{2\pi}\).

Step 3: Calculate the Average Displacement

Average displacement, as found from Step 1 in one complete cycle, is zero because the starting and ending points are the same in periodic motion.

Step 4: Calculate the Average Distance from Rest

The average distance from rest is the average of absolute values of displacement over one full cycle (integral of \(|x(t)|\) over a period divided by the period). Since simple harmonic motion is symmetric, this is half of the amplitude \(A\): \(\frac{1}{T} \int_0^T |x(t)| dt = \frac{2A}{\pi}\).

Key Concepts

Concept of Average Velocity in Simple Harmonic Motion

In the context of simple harmonic motion, average velocity is an interesting concept. It is defined as the total displacement over the total time period for a motion cycle. In simple harmonic motion (SHM), displacement is periodic, and the system returns to its initial position after one full cycle. This means the net displacement for one full oscillation is zero, resulting in an average velocity of zero.
This idea is crucial in understanding why despite an object moving, its average velocity can be zero over complete cycles. Because the starting and ending positions are the same, it does not matter if the object moved—what matters is where it ended up compared to where it began.

• Average velocity = Total displacement / Total time
• In SHM, total displacement for one cycle = 0
• Thus, average velocity = 0

Understanding average velocity helps clarify how motion operates over longer periods and is foundational for studying systems that return to their starting points.

Understanding Average Speed in Simple Harmonic Motion

Average speed provides a different perspective from average velocity because it looks at the total length of the path traveled without regard to direction. Essentially, it is the total distance traveled divided by the time taken to travel that distance. In simple harmonic motion, the path followed by the object is cyclic, meaning it travels to the peak amplitude and returns back.

Imagine a merry-go-round: it swings to and fro between two extremes, hence in one cycle, it travels from the center to one extreme, back through the center to the opposite extreme, and returns again to the center. This makes a total travel distance of four times the amplitude. For simple harmonic motion, the average speed can be calculated as follows:

• Total distance = 4 times the amplitude (4A)
• Time for one cycle = Period, \( T = \frac{2\pi}{\omega} \)
• Average speed = \( \frac{4A}{T} = \frac{4A\omega}{2\pi} \)

This calculation shows the average speed considers the entirety of the motion's journey, providing insights into the dynamics of motion within the set period.

Exploring Angular Frequency in Simple Harmonic Motion

Angular frequency is a fundamental component in the description of simple harmonic motion, denoted in equations by \( \omega \). It characterizes how quickly an object oscillates within a given time and is linked directly with the period of oscillation. The concept is related to the circular motion, where traveling around a circle involves completing cycles, just as oscillations do in SHM.

The angular frequency is mathematically expressed as \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of the oscillation. This relationship highlights the direct connection between how often oscillations happen (frequency) and how long each oscillation takes (period).

• \( \omega \) = Angular frequency
• \( T \) = Period of one complete oscillation
• \( \omega = \frac{2\pi}{T} \)

Understanding angular frequency is essential for predicting the behavior of oscillating systems and analyzing how different factors influence the speed and nature of these rhythms. It essentially sets the tempo for oscillatory systems, like a metronome in music.