To measure the magnitude of the acceleration due to gravity g in an unorthodox manner, a student places a ball bearing on the concave side of a flexible speaker cone ( Figure 1). The speaker cone acts as a simple harmonic oscillator whose amplitude is A and whose frequency f can be varied. The student can measure both A and f with a strobe light. Take the equation of motion of the oscillator as y(t) = A cos (wt + p), where w 2πf and the y axis points upward. If the ball bearing has mass m, find N (t), the magnitude of the normal force exerted by the speaker cone on the ball bearing as a function of time. Your result should be in terms of A, either f (or w), m, g, a phase angle o, and the constant . ► View Available Hint(s) N (t) = IVE ΑΣΦ ?

You have a metal spring of negligible mass that has spring constant k. The spring is supporting an object of mass m which is oscillating up and down with an angular frequency of w. What happens when a large steady current passes through the spring? O The oscillations gradually slow down until they come to a complete stop O The mass continues to oscillate with a reduced oscillation period O The mass continues to oscillate with an increased oscillation period O The mass continues to oscillate with the same oscillation period

In the arrangement shown in the figure below, an object of mass m = 2.0 kg hangs from a cord around a light pulley. The length of the cord between point P and the pulley is L = 2.0 m. (Ignore the mass of the vertical section of the cord.) The left end of a horizontal cord of density ? is connected to a vibrator at point P. A distance L from point P, the cord goes over a pulley and hangs down. A block of mass m connects to the hanging end of the cord. The vibrator causes the portion of cord between point P and the pulley to oscillate such that standing waves are generated. (a) When the vibrator is set to a frequency of 130 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord? kg/m(b) How many loops (if any) will result if m is changed to 72.0 kg? (Enter 0 if no loops form.) loops(c) How many loops (if any) will result if m is changed to 14 kg? (Enter 0 if no loops form.) loops

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 + ...

Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v) Prove tha tx(t0)=x(t0+2piw) In other words, the solution has the same value at time:t0 and at time:t0+2piw regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.

An electron undergoes simple harmonic motion with the acceleration shown below: ax(t)=−amaxsin(2t/T)  with amax=5839 ms2  and T=316 seconds. Assuming that the only motion is oscillatory (ignoring overall translation), what is the maximum speed of the electron? What is the amplitude of the electron's position?

Use the following transformation to solve the linear harmonic oscillator problem: Q = p + iaq,  P = (p − iaq) / (2ia)

A simple device to measure projectile speed consists of a block of ballistic material of mass M attached to a wall by a spring (of negligible mass; we don't know its spring constant) and resting on a flat, frictionless surface. The device is impacted by a projectile of mass m, which embeds itself in the material, after which the two move together in simple harmonic motion with amplitude A and period T. a) During the collision for this system, is energy conserved? Is momentum in the horizontal direction conserved? b) After the collision, is energy conserved? What about momentum? c) What was the initial velocity v of the projectile before the collision (again in terms of m, M, A, and T)? d) Compute v for m = 5 g, M = 1 kg, A = 5 cm, and T = 0.201 s.

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) =