The equation of motion for a damped harmonic oscillator is s(t) = Ae^(−kt) sin(ωt + δ),where A, k, ω, δ are constants. (This represents, for example, the position of springrelative to its rest position if it is restricted from freely oscillating as it normally would). (a) Find the velocity of the oscillator at any time t. (b) At what time(s) is the oscillator stopped?

A mass of 55 grams stretches a spring by 8 cm. (Note that this means the forces balance, and thus mg = kx where m = 55 grams is mass, g = 981 cm/s is acceleration due to gravity, k is the spring constant, and x = 8 cm is the displacement.) The mass is set in motion from this equilibrium position with an initial downward velocity of 23 cm/s, and there is no damping. Find the position u (in cm) of the mass at any time t (in s). (Assume that position is measured upward from the equilibrium position.) u(t) Find the frequency (in radians per second), period (in seconds), and amplitude (in cm) of the motion. Frequency is Period is Amplitude is

A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is 2π ph/g, where h is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced water

Consider a simple harmonic oscillator with natural frequency, w = 1 and the displacement from equilibrium of the mass is denoted by x. At t=0, the mass is released from x=1m with speed 2m/s. What is the magnitude of x at time, t= s?

Problem 3: The motion of critically damped and overdamped oscillator systems is hardly "oscillatory". (a) To illustrate this, prove that a critically damped oscillator passes through the originx = 0 at most once, and determine the relationship between the initial conditions To and vo that is required for the oscillator to pass through the origin. (b) Do the same thing for the overdamped oscillator.

A mass weighing 5 lb stretches a spring 4in. If the mass is pushed upward, contracting the spring a distance of 5 in and then set in motion with a downward velocity of 3 ft/s, and if there is no damping and no other external force on the system, find the position u of the mass at any time t. Determine the frequency (wo), period (T), amplitude (R), and phase (8) of the motion. NOTE: Enter exact answers. Use t as the independent variable. u(t) wo rad/s T = R = ft rad || ||

A meter stick of total length l is pivoted a distance d from one end on a friction-less bearing. The stick is suspended so that it becomes a pendulum. This is called a "physical" pendulum because the mass is distributed over the body of the stick. Assume the total mass is m and the mass density of the stick is uniform. The acceleration of gravity is g. Find T and V as functions of generalized coordinate θ and velocity ˙θ. Do this by considering the stick to be divided into infinitesimal parts of length dl and integrating to find the total kinetic and potential energy. Set up the Lagrangian and find the equation of motion.

A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to √2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 3 ft/s. (Use g = 32 ft/s² for the acceleration due to gravity.) x(t) = Find the time at which the mass attains its extreme displacement from the equilibrium position. t= X x What is the position of the mass at this instant? The extreme displacement is x = 0.39 X feet.

A spring/mass/dashpot system has mass 5 kg, damping constant 70 kg/sec and spring constant 845 kg/sec/sec. Express the ODE for the system in the form a"+ 2px' + wr = 0 Identify the natural (undamped) frequency of the spring: wo 3= (square Hz) Identify the parameter p: (Hz) Now assume that the system has the oscillating forcing function cos(wod) with the same frequendy as the spring's natural frequency. + 14a'+ 169a = cos(wat) Find the general solution.

A mass m = 2.07 kg is attached to a spring of force constant k = 46.9 N/m and set into oscillation on a horizontal frictionless surface by stretching it an amount A = 0.19 m from its equilibrium position and then releasing it. The figure below shows the oscillating mass and the particle on the associated reference circle at some time after its release. The reference circle has a radius A, and the particle traveling on the reference circle has a constant counterclockwise angular speed w, constant tangential speed V = wA, and centripetal acceleration of constant magnitude a. w²A. (a) Determine the following. maximum speed of the oscillating mass m/s magnitude of the maximum acceleration of the oscillating mass +A m/s2 x =-A V = @A magnitude of the maximum force experienced by the oscillating mass N a̟= w³A ot maximum kinetic energy of the oscillating mass maximum elastic potential energy of the spring attached to the mass total energy of the oscillating mass-spring system (c) If the…