Question 2: A point-mass M of mass m is tied to a string which winds around a cylinder like a bobbin. The string is pinned on the cylinder at some point, and P denotes the point at which the string leaves the bobbin and starting from which it is taut. The cylinder has radius a and rotates at a constant rate around the vertical axis of the fixed frame F (which is also the axis of symmetry of the bobbin). The motion takes place in the hori- zontal plane, which is assumed frictionless. See figure. (a) Find a relationship between l, a and expressing that the length of the string is constant (b) Compute the velocity and acceleration of point M with respect to F. (c) Apply LMB in F to the system consisting of the point mass M alone (don't forget to start with a free body diagram!). Deduce a differential equation for l as well as the value of the string's tension in terms of the other parameters. (d) (bonus) Solve the differential equation for l.

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The following​ mass-and-spring system has stiffness matrix K. The system is set in motion from rest (x1′​(0)=x2′​(0)=​0)in its equilibrium position (x1​'(0)=x2​'(0)=​0) with the given external forces F1​(t)=0and F2​(t)=540 cos 4t acting on the masses m1and m2​,respectively. Find the resulting motion of the system and describe it as a superposition of oscillations at three different frequencies.   m1 m2 k1 k2 k3 x1 x2 K= −k1+k2 k2 k2 −k2+k3   m1=​1, m2=2​; k1=1​, k2=2​, k3=2       Find the resulting motion of the system.   x1​(t) =   x2​(t) =   ​(Type exact​ answers, using radicals as​ needed.)