what if a damped harmonic oscillator has a damping constant of beta= 2w, is this overdamping, critical damping, or underdamping? please explain why?
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what if a damped harmonic oscillator has a damping constant of beta= 2w, is this overdamping, critical damping, or underdamping? please explain why?
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- A block of mass m = 2 kg is attached a spring of force constant k = 500 N/m as shown in the figure below.The block is initially at the equilibrium point at x = 0 m where the spring is at its natural length. Then theblock is set into a simple harmonic oscillation with an initial velocity 2.5 m/s at x = 0 cm towards right. Thehorizontal surface is frictionless. a) What is the period of block’s oscillation? b) Find the amplitude A of the oscillation, which is the farthest length spring is stretched to. c) Please represent block’s motion with the displacement vs. time function x(t) and draw the motion graphx(t) for at least one periodic cycle. Note, please mark the amplitude and period in the motion graph.Assume the clock starts from when the block is just released. d) Please find out the block’s acceleration when it is at position x = 5cm. e) On the motion graph you draw for part c), please mark with diamonds ♦ where the kinetic energy of theblock is totally transferred to the spring…Consider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite size ball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillati T = 2π The moment of inertia of the ball about an axis through the center of the ball is Here, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant of acceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball. Note, this pre-lab asks you to do some algebra, and may be a bit tricky. I mgh Iball = / mr² T4.26. Maximum speed * A critically damped oscillator with natural frequency w starts out at position xo > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the origin?
- 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.In figure below, a string, tied to a sinusoidal vibrator at P and running over a support at Q, is stretched by a block of mass m. The separation L between P and Q is 1.2 m, the linear density of the string is 1.6 g/m, and the frequency f of the vibrator is fixed at 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q. What mass m allows the vibrator to set up the fourth harmonic on the string?Question 1. Find the steady state solution of the forced Mass-Spring-Damper with the following parameters undergoing forcing function F(t). 3 kg, c = 22 Ns/m, k = 493 N/m, F(t) = 21 cos(6t) in the form of (t) = A cos(6t – 8) Enter your answers for A and to four decimal places in the appropriate boxes below: m = A: d:
- In this limit, what is the effective spring constant k for the pendulum in the limit of small angles? In this limit, what is the effective angular frequency ω of a pendulum in the limit of small angles?Why is it only possible to produce the odd harmonics in a system with oneopen end and one closed end.A block of m mass is supported by two identical parallel vertical springs, each with spring stiffness constant k. What will be the freuency of vertical oscillation? Answer in terms of variable k, m and appropiate constants.
- a) Sketch the trajectory of a simple harmonic oscillator over the course of one period on the "phase space" plane. The y-axis is the velocity, rescaled by the square root of half of the mass. The x-axis is the position, rescaled by the square root of half of the spring constant. Explain the trajectory on subsequent periods.b) Sketch the trajectory on the same plane for a damped harmonic oscillator over the course of multiple periods.1(a) A damped simple harmonic oscillator has mass 2.0 kg, spring constant 50 N/m, and mechanical resistance 8.0 kg/s. The mass is initially released from rest with displacement 0.30 m from equilibrium. Determine the displacement x(t) as a function of time without assuming weak dissipation. Numerically compute all quantities. (b) The time for transients to become negligible is typically taken to be 5t, where the time constant t is the time required for the amplitude to decay to e-1 of its initial value. Taking the displacement amplitude to be approximately A = 0.30 m, (which holds for weak damping), determine the amplitude at time 5t. = Xoe¬Bt where xoA counter-rotating eccentric mass exciter consisting of two rotating 14-oz weights describing circles of 6-in. radius at the same speed but in opposite senses is placed on a machine element to induce a steady-state vibration of the element and to determine some of the dynamic characteristics of the element. At a speed of 1200 rpm, a stroboscope shows the eccentric masses to be exactly under their respective axes of rotation and the element to be passing through its position of static equilibrium. Knowing that the amplitude of the motion of the element at that speed is 0.6 in. and that the total weight of the system is 300 lb, determine (a ) the combined spring constant k (b ) the damping factor c/cc.