{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

class7a

# class7a - U =-GMm r(77 and in our case of the particle U x...

This preview shows page 1. Sign up to view the full content.

example: A particle of mass m can move under the gravitational attraction of two other particles of equal mass M , fixed at the points ( ± a, 0 , 0). Show that the origin O is a position of equilibrium, but that it is not stable. solution: When the particle is at O , the total force on it is zero, so the origin is indeed an equilibrium position. Whether it is stable or unstable is determined by whether the potential energy function U ( x, y, z ) has a minimum at O . Consider the masses M to be on the x -axis: then F = GMm r 2 (76)
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: U =-GMm r (77) and in our case of the particle U ( x, , 0) =-GMm a-x-GMm a + x =-2 amMG a 2-x 2 (78) Di±erentiating gives ∂U ∂x =-2 amMG [-( a 2-x 2 )-2 (-2 x )] =-4 amMGx ( a 2-x 2 ) 2 (79) indeed this is zero when x = 0. Di±erentiating again: ∂ 2 U ∂x 2 =-4 amMG ( a 2-x 2 ) 2-16 amMGx 2 ( a 2-x 2 ) 3 (80) and evaluate at x = 0: ± ∂ 2 U ∂x 2 ² =-4 mMG a 3 (81) which indicates unstable equilibrium. 11...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online