A particle of mass m is moving in a gravitational field with the potential energy U = -a/r, where r is its distance from the center of the field (with mass µ) and a = Gµm > 0 is const. The corresponding radial force is F, = -OU/Or = -a/r2, according to Newton's law of universal gravitation. The orbit of the particle is an ellipse, given in polar coordinates by r = (1) 1+e cos o where p = M²/(ma) = = a(1 – e2), M is its conserved angular momentum, a is the semi-major axis of the ellipse, and e < 1 is its eccentricity. Substitute r given by equation (1) to the equation for the path in a central field: M2 E = + U, 2mr* [Cd5)" + r²]. and show that the conserved energy of the particle is equal to E = 2a
A particle of mass m is moving in a gravitational field with the potential energy U = -a/r, where r is its distance from the center of the field (with mass µ) and a = Gµm > 0 is const. The corresponding radial force is F, = -OU/Or = -a/r2, according to Newton's law of universal gravitation. The orbit of the particle is an ellipse, given in polar coordinates by r = (1) 1+e cos o where p = M²/(ma) = = a(1 – e2), M is its conserved angular momentum, a is the semi-major axis of the ellipse, and e < 1 is its eccentricity. Substitute r given by equation (1) to the equation for the path in a central field: M2 E = + U, 2mr* [Cd5)" + r²]. and show that the conserved energy of the particle is equal to E = 2a
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![A particle of mass m is moving in a gravitational field with the potential energy U = -a/r, where r is
its distance from the center of the field (with mass µ) and a = Gµm > 0 is const. The corresponding
radial force is F, = -OU/Or = -a/r2, according to Newton's law of universal gravitation. The orbit
of the particle is an ellipse, given in polar coordinates by
r =
(1)
1+e cos o
where p =
M² /(ma) =
= a(1 – e2), M is its conserved angular momentum, a is the semi-major axis
of the ellipse, and e < 1 is its eccentricity.
Substitute r given by equation (1) to the equation for the path in a central field:
M²
E =
dr 2
+ U,
2mr+ [(d6) +r]
and show that the conserved energy of the particle is equal to
E =
2a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4eea6def-32b0-498a-b503-194af21091f4%2F5dfcc5d6-634c-49d1-8b56-b0eced53ed07%2Fi1qno9s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A particle of mass m is moving in a gravitational field with the potential energy U = -a/r, where r is
its distance from the center of the field (with mass µ) and a = Gµm > 0 is const. The corresponding
radial force is F, = -OU/Or = -a/r2, according to Newton's law of universal gravitation. The orbit
of the particle is an ellipse, given in polar coordinates by
r =
(1)
1+e cos o
where p =
M² /(ma) =
= a(1 – e2), M is its conserved angular momentum, a is the semi-major axis
of the ellipse, and e < 1 is its eccentricity.
Substitute r given by equation (1) to the equation for the path in a central field:
M²
E =
dr 2
+ U,
2mr+ [(d6) +r]
and show that the conserved energy of the particle is equal to
E =
2a
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