midterm - Use your solution for an ellipse to determine a...

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

View Full Document Right Arrow Icon
Physics 821 Spring Quarter 2007 Dr. Herbst MIDTERM EXAMINATION TUESDAY, MAY 8 8:30 – 10:18 AM 100 POINTS 1) (40 points) The figure below shows a hoop with mass m and radius a rolling on the inside surface of a fixed hoop with center O and radius b>a under the influence of gravity. a) (20 pts) Find the Lagrangian equation of motion for the angle θ . If there is a constraint, discuss whether it is holonomic or non-holonomic. b) (10 pts) Obtain an expression for the time derivative of θ if the hoop starts from rest at an arbitrary angle. c) (10 pts) Determine what, if any, are the constants of the motion. 2) (30 points) a) (20 pts) A bound particle of mass m moves under the attractive potential V = -k/r . Determine the apsidal distances (minimum and maximum r ) in terms of the total energy E and the angular momentum l . Under what circumstances are there two valid solutions?
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Use your solution for an ellipse to determine a formula for the energy in terms of the semi-major axis a . b) (10 pts) Consider scattering by a central force V(r). Determine an expression for the time as a function of radial distance r assuming that t=0 pertains to the distance of closest approach, 3) (30 points) a) (20 pts) Find the principal moments of inertia and the principal axes for the following system: I xx = a I yy = a I zz = b I xy = c ( c < a ) I xz = I yz = 0 b) (10 pts) Express the unit vectors in the principal axis system in terms of the unit vectors in the initial axis system by a rotation matrix with specific angles. Some Basic Equations d dt " L q # L q = Q k = " # $ % & f q k h = q j j " L q j $ L dh dt = " L t l = mr 2 " V = V + l 2 2 mr 2 l = mvb T = m 2 r 2 + r 2 2 ( ) = $ 2 %...
View Full Document

Page1 / 2

midterm - Use your solution for an ellipse to determine a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online