Three satellites orbit a planet of radius R, as shown in FIGURE EX13.24. Satellites S1 and S3 have mass m. Satellite S2 has mass 2m. Satellite S1 orbits in 250 minutes and the force on S1 is 10,000 N. What is the kinetic-energy ratio for K1 / K3 for S1 and S3?

Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 24c

Chapter 13, Problem 24c

Three satellites orbit a planet of radius R, as shown in FIGURE EX13.24. Satellites S₁ and S₃ have mass m. Satellite S₂ has mass 2m. Satellite S₁ orbits in 250 minutes and the force on S₁ is 10,000 N. What is the kinetic-energy ratio for K₁ / K₃for S₁ and S₃?

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Step 1: Recall the formula for kinetic energy, which is given by \( K = \frac{1}{2} m v^2 \), where \( m \) is the mass of the satellite and \( v \) is its orbital velocity. To find the ratio \( \frac{K_1}{K_3} \), we need to compare the kinetic energies of satellites \( S_1 \) and \( S_3 \).

Step 2: The orbital velocity \( v \) of a satellite is determined by the gravitational force acting as the centripetal force. Using \( F = \frac{G M m}{r^2} \) and \( F = \frac{m v^2}{r} \), equate \( \frac{G M m}{r^2} = \frac{m v^2}{r} \) to solve for \( v \): \( v = \sqrt{\frac{G M}{r}} \).

Step 3: Note that the radius \( r \) of the orbit is proportional to the orbital period \( T \) squared, according to Kepler's third law: \( T^2 \propto r^3 \). Since \( S_1 \) and \( S_3 \) have different orbital periods, their orbital radii will differ. Use \( r \propto T^{2/3} \) to express the relationship between \( r_1 \) and \( r_3 \).

Step 4: Substitute \( r \) into the velocity formula \( v = \sqrt{\frac{G M}{r}} \) for both \( S_1 \) and \( S_3 \). Since \( r \propto T^{2/3} \), the velocities \( v_1 \) and \( v_3 \) will depend on the orbital periods \( T_1 \) and \( T_3 \).

Step 5: Finally, calculate the ratio \( \frac{K_1}{K_3} \) using \( \frac{K_1}{K_3} = \frac{\frac{1}{2} m v_1^2}{\frac{1}{2} m v_3^2} = \frac{v_1^2}{v_3^2} \). Substitute the expressions for \( v_1 \) and \( v_3 \) derived earlier to find the ratio in terms of \( T_1 \) and \( T_3 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. In the context of satellites, this force provides the necessary centripetal force for circular motion.

Kinetic Energy in Orbital Motion

Kinetic energy (K) of an object in motion is given by the formula K = 1/2 mv², where m is the mass and v is the velocity. For satellites in orbit, their velocity is determined by the balance between gravitational force and centripetal force. The kinetic energy of each satellite can be compared to understand their motion and energy dynamics.

Orbital Period and Velocity Relationship

The orbital period (T) of a satellite is the time it takes to complete one full orbit around a planet. The relationship between the orbital period and the radius of the orbit is given by Kepler's third law, which states that T² is proportional to r³, where r is the radius of the orbit. This relationship helps determine the velocity of the satellites, which is crucial for calculating their kinetic energies.

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