Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Write the equation, and then determine the amplitude, period, and frequency of the simple harmonic motion of a particle moving uniformly around a circle of radius 2 units, with angular speed (a) 2 radians per second and (b) 4 radians per second.

Step-by-Step Solution

Answer

Amplitude: 2 units; Case (a) Period: \(\pi\), Frequency: \(\frac{1}{\pi}\); Case (b) Period: \(\frac{\pi}{2}\), Frequency: \(\frac{2}{\pi}\).

Step 1: Equation of Motion

For a particle moving with simple harmonic motion, the equation is given by \( x(t) = A \cos(\omega t) \), where \( A \) is the amplitude and \( \omega \) is the angular speed. In our cases, \( A = 2 \) units.

Step 2: Angular Speed and Amplitude

For both scenarios, the radius is 2 units, which means \( A = 2 \). The angular speed \( \omega \) is given as 2 radians/sec for case (a) and 4 radians/sec for case (b).

Step 3: Period Calculation

The period \( T \) of simple harmonic motion is given by \( T = \frac{2\pi}{\omega} \). For case (a), \( T = \frac{2\pi}{2} = \pi \) seconds. For case (b), \( T = \frac{2\pi}{4} = \frac{\pi}{2} \) seconds.

Step 4: Frequency Calculation

The frequency \( f \) is given by the inverse of the period \( f = \frac{1}{T} \). For case (a), \( f = \frac{1}{\pi} \) Hz. For case (b), \( f = \frac{2}{\pi} \) Hz.

Key Concepts

Amplitude

Amplitude, in the context of simple harmonic motion, describes how far the particle moves from its central position during its motion. It's essentially the 'radius' of motion. Think of amplitude like the maximum stretch of a spring or the height of a pendulum swing. It tells us how far the particle travels from its starting or rest position.

• In our exercise, the amplitude is 2 units.
• This is consistent for both scenarios, regardless of angular speed.

Amplitude is a scalar value, meaning it only has magnitude and no direction. It's a crucial component because it represents the energy in the system — a larger amplitude means more energy is required to reach the peak positions in the oscillation.

Period

The period of a simple harmonic motion is the time it takes for the particle to complete one full oscillation. It's an essential measure because it helps determine how fast the particle is moving through its motion. The period is represented by the letter \( T \) and is given by the formula:
\[T = \frac{2\pi}{\omega}\]

• For angular speed \( \omega = 2 \) rad/s, the period \( T = \pi \) seconds.
• For \( \omega = 4 \) rad/s, the period \( T = \frac{\pi}{2} \) seconds.

Period is inversely proportional to the angular speed: as the speed increases, the period decreases. This relationship is intuitive — faster motion leads to quicker completion of cycles, reducing the time, or period, for each cycle.

Frequency

Frequency refers to how often the particle completes its full motion cycle in a second, measured in hertz (Hz). It's the inverse of the period, highlighting the close relationship between these two concepts. The formula for frequency \( f \) is given by:
\[f = \frac{1}{T}\]

• For \( T = \pi \), the frequency \( f = \frac{1}{\pi} \) Hz.
• For \( T = \frac{\pi}{2} \), the frequency \( f = \frac{2}{\pi} \) Hz.

Frequency shows how many oscillations occur within a unit of time. More cycles per second imply a higher frequency, which occurs when the angular speed is higher. It gives us an easy way to understand how repetitive or "fast" the motion is.