Show that f = pSinwt-mwqCoswt is a constant of motion for the one-dimensional harmonic oscillator.
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Show that f = pSinwt-mwqCoswt is a constant of motion for the one-dimensional harmonic oscillator.
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- A point P in simple harmonic motion has a period of 3 sec- onds and an amplitude of 5 centimeters. Express the motion of P by means of an equation of the form d = a cos wt.Determine the theoretical equation for the dependence of Period of the Physical Pendulum to location of the pivot point for a solid bar of length L. This will be T(x) - Period as a function of pivot point position. Mathematically determine minima of the function T(x).Show that the function x(t) = A cos ω1t oscillates with a frequency ν = ω1/2π. What is the frequency of oscillation of the square of this function, y(t) = [A cos ω1t]2? Show that y(t) can also be written as y(t) = B cos ω2t + C and find the constants B, C, and ω2 in terms of A and ω1
- A point P in simple harmonic motion has a frequency of 2 ocillation per minute and an amplitude of 4 feet. Express the motion of P by means of an equation of the form d = a sin wt.How many nodes are there in the wavefunction of a harmonic oscillator with (i) v = 5; (ii) v = 35?A mass of 0.38 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = (0.26 m)cos[(16 rad/s)t]. Determine the following. %3D (a) amplitude of oscillation for the oscillating mass How does the amplitude of oscillation compare to the magnitude of the maximum displacement from equilibrium? m (b) force constant for the spring N/m (c) position of the mass after it has been oscillating for one half a period m (d) position of the mass one-third of a period after it has been released (e) time it takes the mass to get to the position x = -0.10 m after it has been released
- Consider a mass m attached to a spring with natural length 7, hanging vertically under the action of gravity mgk (where the unit vector k is pointing downwards) and a constant friction force F =-Fok. (a) Find the equilibrium point of the mass, write the equation of motion, and show that the motion of the particle is governed by the fundamental equation of simple harmonic motion. (b) Assume the particle is released from the spring when it has heighth above ground and initial velocity vo. Let y be the height above ground of the particle (note that the orientation of the axis is now opposite of z used in point (a)). Write the equation of motion (under the action of gravity and the friction force F). Solve them for the given initial condition and show that v(y)² = vz+2(g− ¹)(h—−y) m (c) Upon entering the ground (y=0) with velocity v₁, the particle is subject to a constant friction force F₁ where F₁ >0 is a constant. Calculate the distance d travelled by the particle into the ground in…A particle oscillates according to x = Acos(ωt + δ) Determine the phase constant δ if the particle starts from x0 = −A.A simple harmonic oscillator's velocity is given by vy(t) = (0.950 m/s)sin(10.6t – 6.15). Find the oscillator's position, velocity, and acceleration at each of the following times. (Include the sign of the value in your answer.) (a) t = 0 position velocity acceleration (b) position velocity acceleration t = 0.500 s (c) t = 2.00 s position velocity acceleration m m/s m/s² m m/s m/s² m m/s m/s²
- The equation for the angular position of a pendulum oscillating at small angles if given by theta(t)=(.15rad)cos(4.4t+.32) Find the angular frequency in rad/s?Use the following transformation to solve the linear harmonic oscillator problem: Q = p + iaq, P = (p − iaq) / (2ia)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 xo