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Mobile App Chapter 7: Problem 34
Venus has a radius 0.949 times that of Earth and a mass 0.815 times that of Earth. Its rotation period is 243 days. What is the ratio of Venus's spin angular momentum to that of Earth? Assume that Venus and Earth are uniform spheres.
Short Answer
Expert verified
The ratio of Venus's spin angular momentum to that of Earth is approximately 0.00302.
Step by step solution
01
Understand the Problem
The problem asks for the ratio of Venus's spin angular momentum to that of Earth. We know that the radius and mass of Venus are fractions of Earth's radius and mass respectively, and we are given Venus's rotation period.
02
Recall the Formula for Angular Momentum
The formula for the angular momentum (\textbf{L}) of a rotating sphere is given by: \[ L = I \times \frac{2\pi}{T} \]Where \(I\) is the moment of inertia and \(T\) is the rotation period.
03
Calculate the Moment of Inertia
The moment of inertia for a uniform sphere is given by: \[ I = \frac{2}{5} M R^2 \]Where \(M\) is the mass and \(R\) is the radius of the sphere.
04
Express Venus's Physical Quantities in Terms of Earth's Quantities
Given the radius of Venus \( R_v = 0.949 R_e \) and mass \( M_v = 0.815 M_e \), we express the moment of inertia for Venus in terms of Earth's moments of inertia. So, for Venus: \[ I_v = \frac{2}{5} M_v R_v^2 = \frac{2}{5} (0.815 M_e)(0.949 R_e)^2 = \frac{2}{5} (0.815 M_e)(0.900601 R_e^2) \]
05
Simplify the Expression for Venus's Moment of Inertia
Simplify the expression for Venus's moment of inertia: \[ I_v = \frac{2}{5} (0.815 M_e)(0.900601 R_e^2) = 0.815 \times 0.900601 \times \frac{2}{5} M_e R_e^2 \]\[ I_v \approx 0.734 \times \frac{2}{5} M_e R_e^2 \]Therefore: \[ I_v \approx 0.734 \times I_e \]
06
Compute the Ratio of Angular Momentum
Using the angular momentum formula \( L = I \times \frac{2\pi}{T} \), the ratio of the angular momentum of Venus to that of Earth is: \[ \frac{L_v}{L_e} = \frac{I_v \times \frac{2\pi}{T_v}}{I_e \times \frac{2\pi}{T_e}} \]Since \( I_v = 0.734 I_e \) and \( T_v = 243 \) days and Earth's rotation period \(T_e = 1 \) day, we get: \[ \frac{I_v \times \frac{2\pi}{T_v}}{I_e \times \frac{2\pi}{T_e}} = \frac{0.734 I_e \times \frac{1}{243}}{I_e \times 1} = 0.734 \times \frac{1}{243} \]
07
Calculate the Final Ratio
Completing the calculation gives: \[ \frac{L_v}{L_e} = 0.734 \times \frac{1}{243} \approx 0.00302 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Moment of Inertia
The moment of inertia is a measure of an object's resistance to changes in its rotation. It's similar to mass in linear motion but applies to rotational motion. For a uniform sphere, the moment of inertia (\textbf{I}) is given by the formula: \[ I = \frac{2}{5} M R^2 \] Where
- M is the mass of the sphere,
- R is the radius of the sphere.
Rotation Period
The rotation period (\textbf{T}) is the time it takes for a planet to complete one full rotation on its axis. Earth has a rotation period of 1 day, whereas Venus has a much longer rotation period of 243 days. The angular velocity (\textbf{ω}) for a sphere can be derived from its rotation period using the formula: \[ \frac{2\pi}{T} \] As Venus rotates very slowly compared to Earth (\textbf{T} being much larger), its angular velocity (\textbf{ω}) is lower.
Angular Momentum Ratio
Angular Momentum (\textbf{L}) is a quantity that measures the amount of rotation an object has, taking into account its moment of inertia and angular velocity. For a uniform sphere, angular momentum is given by: \[ L = I \times \frac{2\pi}{T} \] We can compute the ratio of angular momentum between Venus and Earth using this formula. Since we've established that Venus's moment of inertia (\textbf{I\textsubscript{v}}) is 0.734 times that of Earth's and knowing their respective rotation periods, the ratio of their angular momenta becomes: \[ \frac{L_v}{L_e} = \frac{0.734 I_e \times \frac{2\pi}{243}}{I_e \times \frac{2\pi}{1}} \approx 0.00302 \]
Uniform Sphere Approximation
This approximation assumes that both Venus and Earth are perfectly uniform spheres. In reality, planets are not perfectly uniform due to variations in density and composition. However, the uniform sphere model simplifies calculations by allowing us to use the formula for moment of inertia: \[ \frac{2}{5} M R^2 \] Using this assumption, we calculated Venus's moment of inertia and compared it to Earth's.
Planetary Properties
Several physical properties are crucial when studying planets like Venus and Earth:
- Radius (R): The distance from the planet's center to its surface; Venus's radius is 0.949 times that of Earth.
- Mass (M): The amount of matter contained in the planet; Venus's mass is 0.815 times that of Earth.
- Rotation Period (T): The time it takes for the planet to complete one full rotation on its axis; for Venus, this is 243 days.
