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Chapter 13: Problem 68

A spherical planet has uniform density \(\rho .\) Show that the minimum period for a satellite in orbit around it is $$T_{\min }=\sqrt{\frac{3 \pi}{G \rho}}$$ independent of the radius of the planet.

Short Answer

Expert verified
The minimum period for a satellite in orbit around a spherical planet is given by \(T_{\min } = \sqrt{\frac{3 \pi} {G \cdot \rho} }\), per the derived formula. This period is independent of the planet's radius, and it only depends on the gravitational constant and the planet's density.

Step by step solution

01

- Understand Kepler's law of gravitation

Before proceeding with the calculation, understand Kepler's third law of planetary motion, which states that the square of the period of a planet's orbit is proportional to the cube of the semi-major axis of its orbit. In mathematical terms, for a planet with orbital period T and semi-major axis a, we have \(T^{2} \propto a^{3}\). In our problem, the planet has uniform density \(\rho\), so the semi-major axis will essentially be its radius R.
02

- Calculate gravitational force

Using Newton's universal law of gravitation, we know that the force of gravity between two bodies is \(F = G \cdot \frac{M \cdot m}{r^{2}}\), where G is the gravitational constant, M is the mass of the planet, m is the mass of the satellite, and r is the distance between the centers of the two bodies. For a sphere with uniform density and volume \(V = \frac{4}{3} \pi R^{3}\), the mass M can be defined as \(\rho \cdot V\). So the formula for the force becomes \(F = G \cdot \frac{\rho \cdot V \cdot m}{R^{2}}\). The force keeps the satellite in circular motion, so it equals \(m \cdot \frac{v^{2}}{R}\), where v is the velocity of the satellite.
03

- Find the velocity

Combine the force formulas from step 2 to solve for the satellite's velocity: \(v^{2} = G \cdot \frac{\rho \cdot V}{R}\). For a full orbit, the distance covered is the circumference of the circle, which is \(2 \pi R\). The speed of the satellite is the distance over time, so we can express v as \(\frac{2 \pi R}{T}\). Substituting this into the velocity equation from above gives us the period: \(T^{2} = \frac{4 \pi^{2} R^{3}}{G \cdot \rho \cdot V}\). Notice that the \(R^{3}\) on the right-side makes this equation in line with Kepler's third law.
04

- Express period in terms of density

The problem asks us to show the period is independent of the planet's radius. To do this, replace the volume V in the previous equation with its expression in terms of the radius. Substituting \(V = \frac{4}{3} \pi R^{3}\) gives \(T^{2} = \frac{3 \pi}{G \cdot \rho}\). Taking the square root gives the final formula for the minimum period of the orbit: \(T_{\min } = \sqrt{ \frac{3 \pi}{G \cdot \rho}}\). This equation shows that the period is indeed independent of the planet's radius, as it only depends on the gravitational constant and the planet's density.

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Key Concepts

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

Kepler's Third Law
In astronomy, Kepler's Third Law of Planetary Motion is fundamental to understanding orbits. This law tells us how the square of a planet鈥檚 orbital period is proportional to the cube of the semi-major axis of its orbit. So, if we have a semi-major axis \(a\), then \(T^{2} \propto a^{3}\).
For simplification, imagine this semi-major axis as the radius of the planet in question. This concept is critical because, when dealing with uniform density, the representation adjusts but the basic proportional rule remains unaffected.
Try visualizing that increasing the radius of the orbit increases the time it takes to complete an orbit. This relationship lets us apply Kepler's law in practical scenarios like orbiting satellites around spherical planets.
Universal Law of Gravitation
The Universal Law of Gravitation, formulated by Isaac Newton, tells us that every two objects in this universe experience an attractive force between them. This force is given by the formula: \[ F = G \cdot \frac{M \cdot m}{r^2} \] where:
  • \( F \) is the gravitational force between the objects.
  • \( G \) is the gravitational constant.
  • \( M \) and \( m \) are the masses of the two objects.
  • \( r \) is the distance between the centers of the two objects.
This force is what keeps a satellite in orbit around a planet. It鈥檚 fascinating to think that this same law applies to everything from fruit falling from trees to planets orbiting stars.
For our calculation, understanding how this force works helps to derive the satellite's motion in its orbit.
Gravitational Constant
The gravitational constant, represented as \( G \), is a key value in the Universal Law of Gravitation. It is a measure of the strength of gravity, allowing us to calculate gravitational forces, whether for falling apples or planets in distant galaxies.
In our formula, \( G \) serves as a scaling factor that adjusts the strength of the gravitational force according to the involved masses and distance. It鈥檚 usually given the approximate value of \(6.674 \times 10^{-11} \text{N m}^2/\text{kg}^2\).
This constant makes sure that the gravitational force calculated is consistent and universally applicable. It鈥檚 a linchpin in calculations involving Newton's law and Kepler鈥檚 law.
Uniform Density
When we talk about uniform density \( \rho \) for a planet, it means that the planet has the same density at every point. Thus, the mass is evenly distributed throughout the entire body.
This makes calculations easier and uniform. For example, mass \( M \) of such a spherical planet can be calculated easily using its volume \( V = \frac{4}{3} \pi R^3 \) and density \( \rho \), as \( M = \rho \cdot V \).
In our original exercise, this uniform density enables us to derive the orbital period \( T_{\min} \) of a satellite independent of the radius of the planet but dependent solely on \( \rho \) and \( G \). Divorcing the formula from the radius makes sense only due to uniform density simplifying the mathematical process.

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Most popular questions from this chapter

Newton's law of universal gravitation is valid for distances covering an enormous range, but it is thought to fail for very small distances, where the structure of space itself is uncertain. Far smaller than an atomic nucleus, this crossover distance is called the Planck length. It is determined by a combination of the constants \(G, c,\) and \(h\) where \(c\) is the speed of light in vacuum and \(h\) is Planck's constant (introduced in Chapter 11 ) with units of angular momentum. (a) Use dimensional analysis to find a combination of these three universal constants that has units of length. (b) Determine the order of magnitude of the Planck length. You will need to consider noninteger powers of the constants.

Studies of the relationship of the Sun to its galaxy-the Milky Way- -have revealed that the Sun is located near the outer edge of the galactic disk, about 30000 lightyears from the center. The Sun has an orbital speed of approximately \(250 \mathrm{km} / \mathrm{s}\) around the galactic center. (a) What is the period of the Sun's galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy? Suppose that the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?

A student proposes to measure the gravitational constant Gby suspending two spherical objects from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. Draw a free-body diagram of one of the objects. If two \(100.0-\mathrm{kg}\) objects are suspended at the lower ends of cables \(45.00 \mathrm{m}\) long and the cables are attached to the ceiling \(1.000 \mathrm{m}\) apart, what is the separation of the objects?

The center-to-center distance between Earth and Moon is \(384400 \mathrm{km} .\) The Moon completes an orbit in 27.3 days. (a) Determine the Moon's orbital speed. (b) If gravity were switched off, the Moon would move along a straight line tangent to its orbit, as described by Newton's first law. In its actual orbit in \(1.00 \mathrm{s}\), how far does the Moon fall below the tangent line and toward the Earth?

In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant \(G\) uses lead spheres with masses of \(1.50 \mathrm{kg}\) and \(15.0 \mathrm{g}\) whose centers are separated by about \(4.50 \mathrm{cm} .\) Calculate the gravitational force between these spheres, treating each as a particle located at the center of the sphere.
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