Find the set of initial conditions such that each mass oscillates: a) at the first natural frequency; b) at the second natural frequency;

Find the natural frequency of a spring having a mass of m and a stiffness k.

A glider of mass 0.350 kg is placed on a frictionless, horizontal air track. One end of a horizontal spring is attached to the glider, and the other end is attached to the end of the track. When released, the glider oscillates in SHM with frequency 4.15 Hz. Find the period of the motion. Find the angular frequency of the motion. Find the force constant kk of the spring. Find the magnitude of the force that the spring exerts on the glider when the spring is stretched by 0.0200 m.

When spinning at 500 rotations per minute two washing machines produce two sinusoidal forces with same maximum amplitude of 400N, but their maximum amplitudes are separated in time by 0.0125s. The two machines are placed side by side and to check their effect on a suspended laundry floor it is necessary to find their resultant vertical force. Find time-varying function describing the resultant of the two vertical forces including the amplitude and phase shift relative to one of the two excitation forces.

6. A 1kg mass is attached to a spring (with spring constant k = 4 N/m), and the spring itself is attached to the ceiling. If you pull the mass down to stretch the spring past its equilibrium position, when you release the mass and observe its (vertical) position, it's said to undergo simple harmonic motion. AT REST MASS PULLED DOWN wwww Under certain initial conditions, the mass's vertical position (in metres) relative to its equilibrium position at time t, y(t), can be modelled by the equation y(t) = cos(2t) – sin(2t), (Note that y measures how much the spring has been stretched, so y = 1 indicates the mass is Im below its equilibrium position, whereas y = -1 indicates it is 1m above its equilibrium position.) (a) Find expressions for the mass's (vertical) velocity v(t) (relative to its equilibrium position) and the mass's (vertical) acceleration a(t) (relative to its equilibrium position). (b) Is the mass moving toward the ceiling or toward the floor at t= T? Justify your answer with…

A simple harmonic oscillator, of mass and natural frequency , experiences anoscillating driving force . Therefore, its equation of motion is where is its position. Given that at we have , , find thefunction . Describe the solution if is approximately, but not exactly, equal to .

ii. A mass spring system has three springs and mass m. Determine the equivalent spring stiffness, period and frequency of this system if the springs are connected in (a) series and (b) parallel. [Take the stiffness of the springs to be u, x and y and the equivalent stiffness as k]

A particle of mass m attached to a rigid support by a support of constant k. At equilibrium, the spring hangs vertically. This mass-spring system is joined by another identical oscillator, whose spring is hung from the previous mass. Consider only the vertical movement. a)  write the equations of motion of the coupled system b)Compute the normal mode frequencies for one-dimensional vertical oscillations and then show that the ratio of the two normal frequencies is √5+1/√5−1 c)Find the ratio of the amplitudes of the two masses in each separate mode. (You do not need to consider the force of gravity acting on the masses because it is independent of the displacements.)

5. A mass m is attached to both a spring (with given spring constant k) and a dashpot (with damping constant c). The mass is set in motion with initial position xo and velocity vo. A) Find the damped position function x(t) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the formx(t) = ce¬pt cos(wit – a1) B) Find the undamped function u(t) = coe-pt cos(@ot – ao) that would result if the mass on the spring were set in motion with the same initial position and velocity but with the dashpot disconnected. m = 3, c = 30, k = 63, xo = 2, vo = 2

Given an oscillator of mass 2.0kg and spring constant of 180N/m, what is the period without damping?  Use numerical methods to model this oscillator with an additional friction force equal to where c is a positive damping constant.  Using c=5.0, what is the new period of oscillation.  What about for c=10? Assume initial position is 0.2m and initial velocity is zero. Please find the period using the position versus time plot and use the first full cycle of the motion.