Multiple Choice
A car is driving up a hill. What is the change in potential energy of the car when it undergoes a displacement of .
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10. Conservation of Energy
Gravitational Potential Energy
Problem 22
Textbook Question
A binary star system has two stars, each with the same mass as our sun, separated by 1.0 ✕ 1012 m. A comet is very far away and essentially at rest. Slowly but surely, gravity pulls the comet toward the stars. Suppose the comet travels along a trajectory that passes through the midpoint between the two stars. What is the comet's speed at the midpoint?
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Step 1: Understand the problem. The comet starts very far away, essentially at rest, and is pulled by the gravitational forces of two stars of equal mass. At the midpoint between the two stars, the gravitational potential energy of the system is converted into the kinetic energy of the comet. Use the principle of conservation of energy to solve the problem.
Step 2: Write the expression for the total gravitational potential energy at the midpoint. The gravitational potential energy due to one star is given by \( U = -\frac{G M m}{r} \), where \( G \) is the gravitational constant, \( M \) is the mass of the star, \( m \) is the mass of the comet, and \( r \) is the distance between the comet and the star. Since there are two stars, the total potential energy at the midpoint is \( U_{total} = 2 \cdot \left(-\frac{G M m}{r} \right) \).
Step 3: Determine the distance \( r \) from the midpoint to each star. Since the stars are separated by \( 1.0 \times 10^{12} \) m, the distance from the midpoint to each star is half of this value: \( r = \frac{1.0 \times 10^{12}}{2} = 5.0 \times 10^{11} \) m.
Step 4: Apply the conservation of energy principle. The comet starts with zero kinetic energy and gravitational potential energy at infinity. At the midpoint, all the gravitational potential energy is converted into kinetic energy. The kinetic energy of the comet is given by \( K = \frac{1}{2} m v^2 \), where \( v \) is the speed of the comet. Set the total gravitational potential energy equal to the kinetic energy: \( 2 \cdot \left(-\frac{G M m}{r} \right) = \frac{1}{2} m v^2 \).
Step 5: Solve for \( v \), the speed of the comet. Cancel \( m \) from both sides of the equation (since \( m \neq 0 \)), and rearrange to isolate \( v \): \( v = \sqrt{\frac{4 G M}{r}} \). Substitute the known values for \( G \) (\( 6.674 \times 10^{-11} \) N·m²/kg²), \( M \) (mass of the sun, \( 1.989 \times 10^{30} \) kg), and \( r \) (\( 5.0 \times 10^{11} \) m) to calculate \( v \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Gravitational Force
Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. In this scenario, the gravitational pull from both stars will influence the comet's trajectory and speed as it approaches the midpoint.
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Conservation of Energy
The principle of conservation of energy states that the total energy in a closed system remains constant. In the context of the comet, as it moves closer to the stars, its gravitational potential energy decreases while its kinetic energy increases, leading to an increase in speed. This principle allows us to relate the comet's initial potential energy to its speed at the midpoint.
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Orbital Mechanics
Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. It encompasses concepts such as trajectories, velocities, and the effects of multiple gravitational bodies. In this case, understanding how the comet's path is affected by the gravitational fields of both stars is crucial for determining its speed at the midpoint.
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