A cylindrical cork (area A, length L, density pc) is floating in a liquid of density Pw with only a part of its length L submerged in the liquid. If the cork is pushed down by a small distance x, and then let go, the cork starts bobbing up and down in a simple harmonic motion. We wish to find the frequency f at which the cork is moving up and down. Let d be the depth of the cork at a certain time t (see figure) and let do be the depth when the cork is in equilibrium (i.e. not bobbing). L d (a) Draw a free body diagram of the cork when it's momentarily at a depth of d < do. Define "up" as the positive direction.

A small piece of rounded stone perforated for threading is sliding on a wire which is hanging freely in the shape of a cosh function. Assume the stone moves with no friction. For very small displacements in x, what is the frequency of oscillation? You may use for the height of the wire and thus the height of the stone as a function of x the Taylor series expansion of cosh. G(x) =1 + x2/2  + x4/24 + ...

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A spring with spring constant k= 7 N/m is horizontal and has one end attached to a wall and the other end attached to a M = 4 kg mass. Suppose that the friction of the mass with the floor (i.e., the damping constant) is D = 1 N · s/m, and the forcing function is F(t) = 2 sin(4t). a. Find the long-term motion of the attached mass if initially the mass is at rest at the equilibrium position. That is, what remains of the solution after all exponentially decaying terms have effectively reached zero. Long-term motion: æ(t) = b. Find the long-term motion of the attached mass if initially the mass is pulled 0.1 metres away from the equilibrium position and is released. Long-term motion: æ(t) =

A horizontal spring mass system oscillates on a frictionless plane.  At time t=0, it is moving left at position x=9 cm.  It has velocity v=0, at positions x=0 cm and 12 cm, and completes one full cycle in 2 seconds.  Write the position and velocity kinematic equations for this oscillating system, including the phase constant.