Download 1. A particle of mass m is projected vertically upward at z=O, with an

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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.
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%)
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4. A 12kg uniform rectangular block with mass center 0 is shown below. At this moment, point 0 has a
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