tailieunhanh - Advanced Engineering Mathematics 2011 Part 14

Tham khảo tài liệu 'advanced engineering mathematics 2011 part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Vector Calculus 509 Example Foucault pendulum One of the great experiments of mid-nineteenth century physics was the demonstration by J. B. L. Foucault 1819-1868 in 1851 of the earth s rotation by designing a spherical pendulum supported by a long wire that essentially swings in an nonaccelerating coordinate system. This problem demonstrates many of the fundamental concepts of vector calculus. The total force1 acting on the bob of the pendulum is F T mG where T is the tension in the pendulum and G is the gravitational attraction per unit mass. Using Newton s second law d2r T S G dt inertial m where r is the position vector from a fixed point in an inertial coordinate system to the bob. This system is inconvenient because we live in a rotating coordinate system. Employing the conventional geographic coordinate system 2 becomes 2ĩ _ dr _ . _ T 2ÍÌ X Í1 X n X r -5 G dt dt TĨĨ where fi is the angular rotation vector of the earth and r now denotes a position vector in the rotating reference system with its origin at the center of the earth and terminal point at the bob. If we define the gravity vector g G fix fixr then the dynamical equation is d2r dr T g 2Bx i g at2 at m where the second term on the left side of is called the Coriolis force. Because the equation is linear let us break the position vector r into two separate vectors ro and n where r ro rj. The vector To extends from the center of the earth to the pendulum s point of support and ti extends from the support point to the bob. Because ro is a constant in the geographic system T 1 From Broxmeyer c. 1960 Foucault pendulum effect in a Schuler-tuned system. J. Aerosp. Sci. 27 343-347 with permission. 2 For the derivation see Marion J. B. 1965 Classical Dynamics of Particles and Systems Academic Press New York Sections . 510 Advanced Engineering Mathematics If the length of the pendulum is L then for small oscillations ri zi yj Lk and the equations of .