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Chapter 7: Problem 45

Two satellites are in circular orbits around the Earth. Satellite A is at an altitude equal to the Earth's radius, while satellite \(B\) is at an altitude equal to twice the Earth's radius. What is the ratio of their periods, \(T_{\mathrm{B}} / T_{\mathrm{A}}\) ?

Short Answer

Expert verified
The ratio of their periods, \(T_{B} / T_{A}\) is approximately \(1.837\).

Step by step solution

01

Set up the relation for Kepler's Third Law

Kepler's Third Law states that the square of the orbital period (\(T\)) of a satellite is directly proportional to the cube of the semi-major axis (\(r\)) of its orbit. This can be mathematically expressed as: \(T^2 \propto r^3 \) or \(T^2 = k \cdot r^3\) where \(k\) is the constant of proportionality.
02

Define the satellite radius and periods

For this problem, the radius of the orbit, \(r\), is equal to the Earth's radius plus the altitude of the satellite. Let's denote the Earth's radius as \(R\). Therefore, for satellite \(A\) (radius \(r_A\)) and satellite \(B\) (radius \(r_B\)) we have: \(r_A = R + R = 2R\), \(r_B = R + 2R = 3R\). Regarding the periods \(T_A\) and \(T_B\), \(T_A\) is the period of satellite \(A\) and \(T_B\) is the period of satellite \(B\). The problem asks for the ratio \(T_{\mathrm{B}} / T_{\mathrm{A}}\).
03

Apply Kepler's Third Law

Applying Kepler's Third Law, we can write: \(T_A^2 = k \cdot (2R)^3\) and \(T_B^2 = k \cdot (3R)^3\). Taking the ratio of these two equations gives us \(T_B^2 / T_A^2 = (3R)^3 / (2R)^3 = (3/2)^3 =27/8\). Since we're looking for the ratio \(T_B / T_A = \sqrt{27/8}\)
04

Simplify the Result

To simplify the ratio, \(\sqrt{27/8} = \sqrt{3.375} \approx 1.837 \) which is the ratio of the periods.

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Key Concepts

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

Orbital Period
The orbital period is the time it takes for a satellite to make a complete orbit around the Earth. This period is crucial in understanding the motion of satellites and can be calculated using Kepler's Third Law.
According to Kepler's Third Law, the square of the orbital period (T) is proportional to the cube of the semi-major axis (r).
  • Mathematically, this is expressed as: \(T^2 = k \cdot r^3\).
  • The constant \(k\) represents the gravitational characteristics of the system.
  • For Earth orbiting satellites, this constant encompasses the planet's gravitational pull.
For satellites like those in the original exercise, knowing the altitude and Earth's radius allows us to compute this period.
Circular Orbits
Satellites orbiting Earth often move in circular orbits due to simplified calculations and stable dynamics.
In circular orbits, the distance from the center of the Earth to the satellite is constant.
  • For a perfect circular orbit, the orbit path is a circle with Earth at the center.
  • The radius of a circular orbit ( r ) is the sum of Earth's radius and the satellite's altitude.
Using these rules, we determine the orbital path and its properties by understanding the balance between gravity and the satellite's velocity.
Satellite Altitude
Altitude refers to the distance of the satellite above Earth's surface, directly influencing orbital dynamics. Calculating this altitude is key to determining orbital characteristics.
  • A satellite's total distance from the Earth's center is its altitude plus Earth's radius.
  • In the exercise, if a satellite's altitude equals Earth's radius, the total distance is twice Earth's radius.
  • This total distance dictates the orbital period, as we computed using Kepler's Third Law.
Thus, satellite altitude is not only a measure of height but also a defining factor in orbital calculations.
Earth's Radius
Understanding Earth's radius is fundamental in satellite calculations. It serves as a baseline for determining orbits. When discussing satellite motion, the radius is used as part of the calculations for distance and orbital parameters.
The Earth's radius is typically about 6,371 kilometers, providing the foundation for creating distance equations in orbital mechanics.
  • As a constant, it helps in calculating the total distance to satellites, by adding satellite altitude.
  • For higher altitudes, Earth鈥檚 radius remains unchanged but significantly influences period calculations.
This constant presence of Earth's radius reveals its importance in all satellite-related physics equations.

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Most popular questions from this chapter

Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

\(\mathrm{}\) A \(40.0-\mathrm{kg}\) child swings in a swing supported by two chains, each \(3.00 \mathrm{~m}\) long. The tension in each chain at the lowest point is \(350 \mathrm{~N}\). Find (a) the child's speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)

A satellite has a mass of \(100 \mathrm{~kg}\) and is located at \(2.00 \times\) \(10^{6} \mathrm{~m}\) above the surface of Earth. (a) What is the potential energy associated with the satellite at this location? (b) What is the magnitude of the gravitational force on the satellite?

An electric motor rotating a workshop grinding wheel at a rate of \(1.00 \times 10^{2} \mathrm{rev} / \mathrm{min}\) is switched off. Assume the wheel has a constant negative angular acceleration of magnitude \(2.00 \mathrm{rad} / \mathrm{s}^{2}\). (a) How long does it take for the grinding wheel to stop? (b) Through how many radians has the wheel turned during the interval found in part (a)?

A pail of water is rotated in a vertical circle of radius \(1.00 \mathrm{~m}\). (a) What two external forces act on the water in the pail? (b) Which of the two forces is most important in causing the water to move in a circle? (c) What is the pail's minimum speed at the top of the circle if no water is to spill out? (d) If the pail with the speed found in part (c) were to suddenly disappear at the top of the circle, describe the subsequent motion of the water. Would it differ from the motion of a projectile?
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