Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 67-70: The formula specifies the position of a point \(P\) that is moving harmonically on a vertical axis, where \(t\) is in seconds and \(d\) is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at \(t=0\) ). $$ d=10 \sin 6 \pi t $$

Step-by-Step Solution

Answer

Amplitude: 10 cm, Period: \(\frac{1}{3}\) s, Frequency: 3 Hz. Motion: Completes one cycle in \(\frac{1}{3}\) seconds.

Step 1: Identify the Amplitude

The formula given for the position of point \(P\) is \(d = 10 \sin 6 \pi t\). The amplitude of a harmonic motion is the coefficient in front of the sine function, which is 10 cm. This is the maximum displacement from the equilibrium position.

Step 2: Determine the Period

The period of a sine function \(\sin(\omega t)\) is calculated using \(T = \frac{2\pi}{\omega}\), where \(\omega\) is the angular frequency. Here, \(\omega = 6\pi\), so the period \(T = \frac{2\pi}{6\pi} = \frac{1}{3}\) seconds.

Step 3: Calculate the Frequency

The frequency \(f\) is the reciprocal of the period. Therefore, \(f = \frac{1}{T} = 3\) Hz. This represents the number of complete oscillations made per second.

Step 4: Describe the Motion

Starting at \(t=0\), the point \(P\) begins at the equilibrium position since \(d=0\) when \(\sin(0)=0\). It moves to a maximum displacement of +10 cm in one direction, returns through zero to -10 cm in the opposite direction, and then back to zero, completing one cycle. The full cycle takes \(\frac{1}{3}\) seconds.

Key Concepts

Understanding Amplitude in Harmonic Motion

In harmonic motion, the amplitude represents the maximum displacement from the equilibrium position. Imagine swinging on a swing; the farthest point you reach from where you started is the amplitude. This concept is crucial as it shows how "big" or "intense" the motion is.
In the formula for the point motion, \(d = 10 \sin 6 \pi t\), the amplitude is 10 centimeters. The formula tells us that point \(P\) can move 10 centimeters above or below its rest position. The starting point remains in the center as the point moves up and down.
The role of amplitude in harmonic motion can be summarized as:

• It indicates the peak value of displacement.
• A larger amplitude means greater energy involved in the motion.
• It remains constant unless external forces modify the system.

Amplitude is a fundamental aspect as it influences the dynamics of systems, from a simple pendulum to complex vibrating strings.

The Period of Harmonic Motion

The period is the time it takes to complete one full oscillation. In simpler terms, it is how long it takes to return to the starting position with the same velocity and direction. If you watch a pendulum, the period is like timing from one side to the other and back again.
In our exercise, the angular frequency \(\omega\) is \(6\pi\). Using the formula for period \(T = \frac{2\pi}{\omega}\), we find:
\[T = \frac{2\pi}{6\pi} = \frac{1}{3}\text{ seconds}\]
This means the motion completes a cycle every \(\frac{1}{3}\) seconds. The period tells you how "fast" or "slow" the oscillation is. Interesting points about period include:

• A smaller period means more quick and fast oscillations.
• The period is independent of amplitude; changing the amplitude does not affect how long one cycle takes.
• It is essential in designing watches, music instruments, and even heart pace monitoring devices.

Understanding the period helps in predicting the timing and behavior of oscillating systems.

Frequency of Oscillation

Frequency measures how often a repeating event occurs per unit time. It indicates the number of complete oscillations in one second. The more frequently something happens, the higher the frequency.
From the exercise, after finding the period \(T = \frac{1}{3}\) seconds, the frequency \(f\) is the reciprocal. So,\[f = \frac{1}{T} = 3\,\text{Hz}\]
Here, \(3\,\text{Hz}\) implies that the point completes 3 cycles every second. The concept of frequency is vastly used in various fields:

• In music, pitch depends heavily on frequency; higher frequencies produce higher pitches.
• In technology, frequency determines capabilities in wireless communication.
• In everyday life, frequency tells us the color of light or the tone of sound.

Understanding frequency gives insights into the underlying energy and behavior of periodic motions.