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lij~: li.tll:nxJ17*~ 102 ~~fl~±Flfm~~~~m 14o *.PJTHJiJJU : JlfJL~x~Ifi~-*pgHJi ~~itf4§ : !lrt.J~ ~~BWI: 0223 • f!J~: 2 1. A particle of mass m is projected vertically upward at z=O, with an initial velocity v0 • Assuming for an upward velocity v 2: 0, there is a square-law drag force bv2, where b is constant. Express vas a function of the vertical displacement and find the maximum height of the particle. (20%) 2. Particles m1=2m and m2 =m can slide without friction on parallel fixed horizontal wires separated by a distance h. A spring of stiffness k and unstressed length h connects the two particles. If m1 has an initial velocity vo, m2 is initially motionless, and the spring is initially unstressed, find: (a) the maximum velocity v2 ofm2; (b) the maximum stretch f5 ofthe spring. (30%) 3. Consider the system shown below, where mass m is a magnetic bar and is sliding in a solenoid that is mounted on a cart of mass M. Let XI and x2 be the absolute positions ofM and m, respectively. A force F is applied to m in the positive x 2 direction if a positive voltage V is applied to the solenoid, and F= J..L V, where ~ is a constant. For simplicity, let ~=1. Suppose friction is negligible. m / a. Defive the governing equation ofthe system. (4%) b. At this moment both M and mare still at the position (x 11 x 2 ) = (0,0). You are asked to design V(t) to move both masses to the position (x 11 x 2 ) = (1,1) in 10 seconds. Please design V(t), or, explain why you cannot do it? (8%) ~~ : 14o IIJJl:nX:~*~ 102 ~~Jliji±~JEm~~~~m *.PJT*J15JU : *IL~::*:~I~j~*~*Jl ~~f4 El : IJrfJ* ~lti\:BWl: 0223 • iil~: 2 4. A 12kg uniform rectangular block with mass center 0 is shown below. At this moment, point 0 has a velocity V= 21 + 3) mjs, and the block is rotating with an angular velocity ro = 1i + lj + lk radjs. a. Find the inertia matrix of the block with respect to the frame (x, y, z). (3%) b. Find the moment of inertia about the axis AB. (7%) c. Find the angular momentum of the block. (3%) d. A force F = lj N is applied at comer A. Find the acceleration at comer C. (15%) e. At this moment, what is the power delivered to the block by the force F? (10%) Unit: m