HW2_prob2_GPS19_G17

# HW2_prob2_GPS19_G17 - Goldstein 1.17(3rd ed 1.19 Because...

This preview shows pages 1–2. Sign up to view the full content.

Goldstein 1.17 (3 rd ed. 1.19) Because the mass point is suspended by a rigid rod of length a , the system has only two degrees of freedom. The two generalized coordinates may be taken to be the polar and azimuthmal angles of a spherical coordinate system centered on the point of attachment of the rigid rod, and we will let the polar ( z ) axis point downwards so that the polar angle θ is positive when the mass point is displaced away from its rest position. The kinetic energy of the mass point is T = m 2 ( · x 2 + · y 2 + · z 2 ) . (1) In the spherical coordinate system we have x = a sin θ cos φ , (2) y = a sin θ sin φ , (3) z = a cos θ , (4) from which it follows that · x = a [ · θ cos θ cos φ · φ sin θ sin φ ] , (5) · y = a [ · θ cos θ sin φ + · φ sin θ cos φ ] , (6) · z = a · θ sin θ . (7) After substituting Eqs.(5)—(7) into Eq.(1) we obtain T = 1 2 ma 2 [ · θ 2 + · φ 2 sin 2 θ ] . (8) Since the z axis points down in our coordinmate system, the potential energy is V = mgz = mga cos θ . (9) Thus, the Lagrangian of the pendulum is

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern