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homework1sol.20100130.4b64aec529df82.58647819

# homework1sol.20100130.4b64aec529df82.58647819 - r 1 sin θ...

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TAM 412, Homework 1, Selected answers Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign 1. The map ξ : P 7 ( x ( P ) ,y ( P )) is onto and one-to one. Also, s 1 = y cos α x cos α q 0 2 = x cos β + y sin β and, consequently, x = q 0 2 cos α s 1 sin β cos ( α β ) y = q 0 2 sin α + s 1 cos β cos ( α β ) provided that α β 6 = ± π / 2 . 2. On the set of con f gurations where ( r 1 , sin θ ) are coordinate functions, A = 1 2 r 2 1 sin θ p 1 sin 2 θ . On this set of con f gurations, the determinant of the corresponding jacobian of the transformation
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Unformatted text preview: ( r 1 , sin θ ) 7 → ( A, sin θ ) is singular when θ = 0 . 3. The jacobian is singular for all values of ( θ , φ , α ) . The triangle with sides of length R and r and hypotenuse given by line segment from origin to particle is a right-angled triangle. The particle moves on the surface of a sphere. 4. When we rotate about the axis by 2 π / 3 the vector e 1 becomes ˜ e 1 = e 2 . 1...
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