Explanation Given: Earth revolves around the sun in an approximately circular orbit with radius r = a , completing a revolution in the time T = 2 π ( a 3 / G M ) 1 / 2 , which is one Earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on Earth is given by G M m / r 2 , where m is the mass of Earth. Therefore, if Earth “stood still,” losing its accordance with Newton’s second law: m d 2 r d t 2 = − G M m r 2 . Approach: The Energy Integral Lemma: Let y ( t ) be a solution to the differential equation y ″ = f ( y ) where f ( y ) is a continuous function that does not depend on y ′ or the independent variable t . Let F ( y ) be an infinite integral of f ( y ) , that is, f ( y ) = d d y F ( y ) Then the quantity, E ( t ) = 1 2 ( y ′ ( t ) ) 2 − F ( y ( t ) ) is constant; that is, d d t E ( t ) = 0 . Calculation: Rearrange the relation, m d 2 r d t 2 = − G M m r 2 . d 2 r d t 2 = − G M r 2 Apply the energy integral lemma, F ( r ) = ∫ − G M r 2 = G M r ( 1 ) Thus, k = 1 2 ( r ′ ( t ) ) 2 − G M r ( t ) … ( 1 ) Apply the initial conditions r ′ ( 0 ) = 0 , r ( 0 ) = a in equation ( 1 ) . k = 0 − G M a = − G M a Substitute − G M a for k in equation ( 1 ) . − G M a = 1 2 ( r ′ ( t ) ) 2 − G M r ( t ) 1 2 ( r ′ ( t ) ) 2 = G M r ( t ) − G M a ( d r d t ) 2 = 2 G M ( 1 r − 1 a ) d r d t = ± 2 G M ( 1 r − 1 a ) As r is decreasing so negative sign will be considered, d r d t = − 2 G M ( 1 r − 1 a ) This implies, − d r 1 r − 1 a = 2 G M d t Integrate on both sides, − ∫ d r ( 1 r − 1 a ) = 2 G M ∫ d t ... ( 2 ) Use the method of substitution, substitute u 2 = 1 r − 1 a . Differentiate u 2 = 1 r − 1 a with respect to u . 2 u d u = − r 2 d r As r 2 = a 2 ( a u 2 + 1 ) 2 , d r = − 2 u a 2 ( a u 2 + 1 ) 2 d u Substitute d r = − 2 u a 2 ( a u 2 + 1 ) 2 d u in equation ( 2 ) . − 2 a 2 ∫ d u ( u 2 a + 1 ) 2 = 2 G M ( t ) + C .... ( 3 ) Again use the method of substitution, substitute u = 1 a tan θ

Fundamentals of Differential Equations [With CDROM] - 7th Edition

7th Edition

ISBN: 9780321410481

Author: Saff, Edward B., Snider, Arthur David, Nagle, R. Kent

Publisher: Addison Wesley

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Chapter 4.8, Problem 17E

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The fraction of the normal year T that would require by Earth to splash into the sun.

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Chapter 4.8, Problem 16E

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Fundamentals of Differential Equations [With CDROM] - 7th Edition

Ch. 4.1 - Verify that for b=0 and Fext(t)=0, equation (3)...Ch. 4.1 - If Fext(t)=0, equation (3) becomes my+by+ky=0. For...Ch. 4.1 - Show that if Fext(t)=0, m=1, k=9, and b=6, then...Ch. 4.1 - Prob. 4E Ch. 4.1 - Verify that the exponentially damped sinusoid...Ch. 4.1 - An external force F(t)=2cos2t is applied to a...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Undamped oscillators that are driven at resonance...

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