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Chapter 12: Problem 45

The estimated amount of zodiacal dust in the Solar System remains constant at approximately \(10^{16}\) kg. Yet zodiacal dust is constantly being swept up by planets or removed by the pressure of sunlight. a. If all the dust disappeared (at a constant rate) over a span of 30,000 years, what would the average production rate, in kilograms per second, have to be to maintain the current content? b. Is this an example of static or dynamic equilibrium? Explain your answer.

Short Answer

Expert verified
a. The production rate is approximately \(1.057 \times 10^{4} \text{ kg/sec}\). b. This is an example of dynamic equilibrium.

Step by step solution

01

- Understand the Given Information

The total amount of zodiacal dust is constant at approximately \(10^{16}\) kg. The rate of disappearance is such that all the dust would be gone in 30,000 years if it were not replenished.
02

- Convert Time to Seconds

Convert 30,000 years to seconds. Use the conversion factors: 1 year = 365.25 days, 1 day = 24 hours, 1 hour = 3600 seconds.\[30,000 \text{ years} \times 365.25 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \]Calculate the total number of seconds.
03

- Calculate Total Seconds

Calculate the total number of seconds.\[30,000 \times 365.25 \times 24 \times 3600 \text{ sec} = 9.46 \times 10^{11} \text{ sec}\]
04

- Determine the Production Rate

Using the total amount of dust and the time period in seconds, calculate the average production rate to maintain the dust content.\[\text{Production rate} = \frac{10^{16} \text{ kg}}{9.46 \times 10^{11} \text{ sec}} \]Solve the division.
05

- Division for Production Rate

Perform the division to find the production rate.\[\text{Production rate} = 10^{16} \text{ kg} \times (1.057 \times 10^{-12} \text{ sec}^{-1}) = 1.057 \times 10^{4} \text{ kg/sec}\]
06

- Determine the Type of Equilibrium

Evaluate if this is an example of static or dynamic equilibrium. Since the dust is constantly being removed and equally replenished, this is an example of dynamic equilibrium.

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Key Concepts

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

mass calculation
Understanding how to handle large masses, like zodiacal dust in the Solar System, is crucial.
We start with a given mass of approximately 1016 kg.
This accounts for all the zodiacal dust currently in the solar system.
Our goal is to maintain this mass, even though it's constantly lost.
To do so, we need to produce the same amount of dust over time.
Think of it as a balance.
The mass is gradually reduced but must be replenished to stay constant.
conversion to seconds
When dealing with astronomical events, large spans of time are common.
Here, we need to convert 30,000 years into seconds for precise calculations.
First, know that 1 year is approximately 365.25 days.
This includes leap years.
Next, each day has 24 hours, each hour has 3600 seconds.
The conversion formula is:
  • 30,000 years × 365.25 days/year
  • × 24 hours/day
  • × 3600 seconds/hour

Multiplying these out gives us a total of 9.46 × 1011 seconds.
This provides a precise time span for our calculations.
production rate
To maintain the dust content, we must replace what is lost.
This is done at a specific production rate.
First, use the total mass and the converted time span:
  • Total mass: 1016 kg
  • Total time: 9.46 × 1011 seconds
The formula to find the production rate is:
  • Production rate = Total mass / Total time
Simplify as follows:
  • Production rate = 1016 kg ÷ 9.46 × 1011 sec
  • Production rate ≈ 1.057 × 104 kg/sec
This rate ensures we constantly replenish the zodiacal dust.
dynamic equilibrium
Dynamic equilibrium means there's a constant replacement happening.
In our case, zodiacal dust is continually lost but also produced.
Unlike static equilibrium, which is unchanging, dynamic equilibrium has ongoing adjustments.
Here, the production of 1.057 × 104 kg/sec balances the dust lost.
Thus, the overall amount stays around 1016 kg.
This dynamic process ensures the Solar System maintains its cosmic dust levels.
It's an equilibrium because additions and losses match perfectly over time.

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