Concept explainers
(a)
The Earth’s orbital speed at aphelion.
(a)
Answer to Problem 13.69AP
The Earth’s orbital speed at aphelion is
Explanation of Solution
The maximum distance from the Earth to the Sun is
By the conservation of
Here,
The angular momentum at perihelion is given as,
Here,
The angular momentum at aphelion is given as,
Here,
Substitute
Substitute
Conclusion:
Therefore, the Earth’s orbital speed at aphelion is
(b)
The kinetic and potential energy of the Earth-Sun system at perihelion.
(b)
Answer to Problem 13.69AP
The kinetic of the Earth-Sun system at perihelion is
Explanation of Solution
Formula to calculate the kinetic energy of the Earth-Sun system at perihelion is,
Substitute
Formula to calculate the potential energy of the Earth-Sun system at perihelion is,
Here,
Substitute
Conclusion:
Therefore, the potential energy of the Earth-Sun system at perihelion is
(c)
The kinetic and potential energy of the Earth-Sun system at aphelion.
(c)
Answer to Problem 13.69AP
The kinetic of the Earth-Sun system at aphelion is
Explanation of Solution
Formula to calculate the kinetic energy of the Earth-Sun system at aphelion is,
Substitute
Formula to calculate the potential energy of the Earth-Sun system at aphelion is,
Here,
Substitute
Conclusion:
Therefore, the potential energy of the Earth-Sun system at aphelion is
(d)
Whether the total energy of the Earth-Sun system constant.
(d)
Answer to Problem 13.69AP
Yes, the total energy of the Earth-Sun system is remains constant.
Explanation of Solution
Formula to calculate the total energy of the Earth-Sun system at aphelion is,
Here,
Substitute
Formula to calculate the total energy of the Earth-Sun system at perihelion is,
Here,
Substitute
Mathematically proved, the sum of kinetic energy and potential energy of the Earth–Sun system at perihelion is identical to the sum of kinetic energy and potential energy of the Earth–Sun system at aphelion. So the total energy of the Earth-Sun system is constant.
Conclusion:
Therefore, yes, the total energy of the Earth-Sun system remains constant.
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