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hw6solution

# hw6solution - Homework 6 Solution 1 Draw the FBDs for the...

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Homework 6 Solution 1. Draw the FBDs for the two blocks: m 1 m 2 x 1 x 2 r 1 1 k x 2 2 k x 2 1 ( 2 ) c x x ± ± F C θ Note that the displacement at the top of the disk is twice the displacement x 1 at the disk’s center. Apply Newton’s 2nd law: M C J C ̈ k 1 x 1 r c x ̇ 2 2 x ̇ 1  2 r J C ̈ F x ma x c x ̇ 2 2 x ̇ 1 k 2 x 2 m 2 x ̈ 2 The moment of inertia of the disk with respect to the point of contact is J C J m 1 r 2 1 2 m 1 r 2 m 1 r 2 3 2 m 1 r 2 From kinematics we have x 1 r The moment equation can then be rewritten as 3 2 m 1 r 2 x ̈ 1 r c x ̇ 2 2 x ̇ 1  2 r k 1 x 1 r 0 After some algebraic simplification, the equations of motion become 3/2 m 1 x ̈ 1 4 cx ̇ 1 2 cx ̇ 2 k 1 x 1 0 m 2 x ̈ 2 2 cx ̇ 1 cx ̇ 2 k 2 x 2 0 Recast the equations of motion in the matrix form: 3 m 1 /2 0 0 m 2 x ̈ 1 x ̈ 2 4 c 2 c 2 c c x ̇ 1 x ̇ 2 k 1 0 0 k 2 x 1 x 2 0 0 2. The FBDs of the rod and the block are drawn:

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