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A simple pendulum suspended in a carriage traveling with a constant acceleration a in the X direction. Find Lagrange's equations of motion. Also find the frequency of their small vibrations.
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- 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 system begins at rest with the given values (3), the system has damped harmonic oscillator and damping constant provided by the equation (1), that is influenced by the eqn (2). find the equation of motion and find the complementary solution of x(t). find all the coefficients and show work pleaseA disk of radius 0.25 meters is attached at its edge to a light (massless) wire of length 0.50 meters to form a physical pendulum. Assuming small amplitude motion, calculate its period of oscillation. For a disk, I = (1/2)MR2 about its center of mass.
- Consider an elastic string of length L == 10 whose ends are held fixed. The string is set in motion from its equilibrium position with an initial velocity ut(x, 0) = g(x). In the following problems, let a = 1 and find the displacement u(x, t) for the given initial velocity g(x). 4x/L, 0 < xone-dimensional crystal assembled from three identical atoms connected to one another and to the walls by identical springs as shown.find the normal modes of given system.plot relative displacements versus time for all three normal modes.A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity. Show that the following Lagrangian gives the desired equation of motion: L = e2γt(1/2 * mq˙2 − V (q))