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### me104-sp08-mt1-Ma-soln

Course: MECHANICAL 104, Spring 2012
School: Berkeley
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Word Count: 295

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(a) 1. A coordinate system attached to car A is a translating system, in which aB = a A + aB / A Observe that ( a B ) t = 3 m/s2 2 v B 144 (a B ) n = = = 1.44 m/s2 100 Thus a B = 3.33 m/s2 64.4 Either from a graphical or analytical solution, a B / A = 5.32 m/s2 = 62.72 (b) A coordinate system attached to B is a rotating system. If a rel is the acceleration of car A as observed from car B, a rel a B / A . It...

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(a) 1. A coordinate system attached to car A is a translating system, in which aB = a A + aB / A Observe that ( a B ) t = 3 m/s2 2 v B 144 (a B ) n = = = 1.44 m/s2 100 Thus a B = 3.33 m/s2 64.4 Either from a graphical or analytical solution, a B / A = 5.32 m/s2 = 62.72 (b) A coordinate system attached to B is a rotating system. If a rel is the acceleration of car A as observed from car B, a rel a B / A . It can be shown from rigid-body kinematics that a rel satisfies & a A = a B + rA / B + ( rA / B ) + 2 v rel + a rel aB = 3.33 m/s2 aB/A 60 64.4 aA = 2 m/s2 2. Let T1 be the tension in the upper string and T2 tension in the lower string. Force balance on each mass gives && (1) 4mg T1 = 4mq1 && 3mg T2 = 3mq3 (2) && (3) mg T2 = mq4 && && && There are five unknowns q1 , q3 , q4 , T1 , and T2 in three equations. However, && && && && && && && && q3 2 q + ( q 4 q 2 ) = 0 q3 + q4 = 2q2 = 2 q1 T1 = 2T2 Upon solution, 1 && q1 = g = 1.40 m/s2 7 q4 q2 q1 T1 T1 4m T2 q3 3m T2 m 3. Let position 1 of the block be its initial position at 150 mm above the springs. Suppose position 2 corresponds to deflection x in the two springs. Between positions 1 and 2, U = T + Vg + Ve where the work done by forces other than gravitational and spring forces is U =0 T = T2 T1 = 0 Define the reference level for measuring potential energy as the level associated with the precompressed springs before impact. Then Vg = mgh2 mgh1 = mg ( x ) mg (0.15) = mg ( x + 0.15) 1 2 1 Ve = 2 kx2 2 kx12 2 2 1 1 = 2 k (0.075 + x ) 2 2 k (0.075) 2 = k [(0.075 + x ) 2 0.0752 ] 2 2 Thus U = T + Vg + Ve mg ( x + 0.15) + k [(0.075 + x ) 2 0.0752 ] = 0 5000 x 2 + 651.9 x 14.715 = 0 x = 0.0196 or 0.150 The additional deflection is x = 19.6 mm. 10 kg Position 1 150 mm Reference x Position 2 k k = 5 kN/m
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