homework3.20100209.4b71a4dbbd35a8.25707887

homework3.20100209.4b71a4dbbd35a8.25707887 - TAM 412,...

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

View Full Document Right Arrow Icon
TAM 412, Homework 3, due February 10, 2010 Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign danko@illinois.edu Please refer to the revised lecture notes on the website for completing this homework assignment. 1. Consider a particle constrained to a plane. Consider the collection C of allowable con f gurations of the particle within a distance R from the point O at the intersection of two orthogonal axes X and Y . Let ξ = ¡ xy ¢ T ,where x and y are the signed distances from O to the corresponding orthogonal projections of the particle onto the X and Y axes. (a) Explain why ( C , ξ ) is a coordinate chart. (b) Explain why ( C , ξ ) is not a global chart. (c) Let ¯ ξ = ¡ στ ¢ T and consider the relationship x = ,y = 1 2 ¡ τ 2 σ 2 ¢ Denote by C 0 the con f guration corresponding to σ = τ =1 . Find a collection C3 C 0 of allowable con f gurations such that ¡ C , ¯ ξ ¢ is a coordinate chart. (d) Explain why there is no global chart with coordinate functions given by σ and τ . (e) Restrict attention to allowable con f gurations that satisfy the equation ( x 1) 2 y =0 Find the largest collection b C 0 of allowable con f gurations such that ³ b C , τ ´ is a coordinate chart. (f) In a single diagram, use Mathematica to graph the coordinate curves corresponding to constant σ for σ between 2 and 2 in as small increments as you feel useful. In the same diagram graph the coordinate curves corresponding to constant τ for τ between 2 and 2 ,aswe l lasthecurve corresponding to the constraint in e. Use the result to explain your answers to d. and e. [Hint: where do the coordinate curves become tangent? Where does the curve in e. become tangent to
Background image of page 1

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

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

Page1 / 3

homework3.20100209.4b71a4dbbd35a8.25707887 - TAM 412,...

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