A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length μ(x) increases uniformly from μ0 at x = 0 to μLat x = L. (a) Find an expression for μ(x) as a function of x over the range 0 ≤ x ≤ L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.

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 + ...

Hooke's law describes a certain light spring of unstretched length 33.0 cm. When one end is attached to the top of a door frame and a 5.00-kg object is hung from the other end, the length of the spring is 48.5 cm. Answer parts a-b.

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) =