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Chapter 5: Problem 37

The variable star Zeta Gemini has a period of 10 days. The average brightness of the star is 3.8 magnitudes, and the maximum variation from the average is 0.2 magnitude. Assuming that the variation in brightness is simple harmonic, find an equation that gives the brightness of the star as a function of time.

Short Answer

Expert verified
The brightness function is \( y(t) = 0.2 \cos\left( \frac{\pi}{5} t \right) + 3.8 \).

Step by step solution

01

Understand Simple Harmonic Motion

The problem states that the brightness variation follows simple harmonic motion. Simple harmonic motion can be described by the equation \( y(t) = A \cos(\omega t + \phi) + D \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, \( \phi \) is the phase shift, and \( D \) is the average value around which the function oscillates.
02

Identify the Amplitude

The maximum variation from the average brightness is given as 0.2 magnitude. This is the amplitude \( A \) of the simple harmonic motion. Hence, \( A = 0.2 \).
03

Determine the Average Brightness

The average brightness given in the problem is 3.8 magnitudes. This is the value around which the brightness oscillates, represented by \( D \) in the equation. Hence, \( D = 3.8 \).
04

Calculate the Angular Frequency

The period \( T \) of the brightness variation is 10 days. The angular frequency \( \omega \) is given by the formula \( \omega = \frac{2\pi}{T} \). Substituting \( T = 10 \), we get \( \omega = \frac{2\pi}{10} = \frac{\pi}{5} \).
05

Set the Phase Shift

The phase shift \( \phi \) is not specified in the problem. For simplicity, we can assume \( \phi = 0 \). In actual applications, the phase shift would be determined by the initial conditions of the system.
06

Formulate the Brightness Equation

Substitute the values \( A = 0.2 \), \( \omega = \frac{\pi}{5} \), \( \phi = 0 \), and \( D = 3.8 \) into the simple harmonic motion equation: \[ y(t) = 0.2 \cos\left( \frac{\pi}{5} t \right) + 3.8 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
In simple harmonic motion (SHM), the amplitude is a crucial concept. It represents the maximum extent of the vibration or oscillation from its mean position. In the context of the star Zeta Gemini, the amplitude signifies how much the brightness varies from its average value. Here, the problem states that the maximum fluctuation in brightness is 0.2 magnitudes. Therefore, the amplitude is this value: 0.2.
This means that the star's brightness can be 0.2 magnitudes brighter or dimmer than its average. Understanding amplitude helps in visualizing the extent of oscillation in systems like pendulums, springs, or stars with varying brightness.
  • Amplitude is the peak value from the mean.
  • In SHM, it defines the max displacement from equilibrium.
  • For Zeta Gemini, it equals 0.2 magnitudes.
Angular Frequency
Angular frequency, denoted by \( \omega \), tells us how fast an object is oscillating in SHM. It's an important parameter because it defines how many oscillations occur in a unit of time.
Mathematically, it is given by:
\[\omega = \frac{2\pi}{T}\]where \( T \) is the period of the oscillation. For Zeta Gemini with a 10-day period, the angular frequency calculates to:
\( \omega = \frac{2\pi}{10} = \frac{\pi}{5} \).
This result indicates how rapidly the brightness changes per unit time. Angular frequency is essential for linking the temporal aspect of oscillations with their physical observations.
  • Counts rotations per unit time.
  • For Zeta Gemini, \( \omega = \frac{\pi}{5} \).
  • Higher \( \omega \) means quicker oscillations.
Phase Shift
In SHM, the phase shift \( \phi \) determines where in its cycle the oscillation begins at time zero. It is an angular measurement, indicating an offset in the starting position of the wave compared to a standard.
Phase shift becomes particularly important when initial conditions differ, but in the Zeta Gemini problem, it's assumed to be 0 for simplicity.
This decision simplifies calculations and assumes the cycle starts from the mean position (when the cosine function is at its peak). However, real-world scenarios might include a phase shift to accurately represent the initial starting point of oscillations.
  • Indicates initial angle or point in the cycle.
  • Assumed 0 for Zeta Gemini.
  • Phase shift allows customization of wave start.
Brightness Equation
The brightness equation in simple harmonic form models how the brightness of Zeta Gemini varies over time. This equation combines all the elements we've discussed: amplitude, angular frequency, and phase shift.
For Zeta Gemini, the brightness equation is as follows:
\[ y(t) = 0.2 \cos\left( \frac{\pi}{5} t \right) + 3.8 \].
This formula allows us to predict brightness at any given time \( t \). Here:
  • \( 0.2 \) is the amplitude.
  • \( \frac{\pi}{5} \) represents angular frequency.
  • \( 3.8 \) is the average brightness.
The equation captures the periodic nature, showing how Zeta Gemini's light varies cyclically due to SHM.

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