1. A pendulum consist of a particle with mass M attached to a massless piece of string of lengthl which is suspended from the bottom of a spring with spring constant k. The spring is guidedwithin a column such that the connection to the pendulum can only move vertically. Referringto the illustration below, let z represent the amount the spring is stretched. Determine a) theLagrangian for the system in terms of the two independent coordinates 0 and z.and b) theequation of motion for just the z-coordinate. Rem: the equation of motion is the diff eq, not thesolution to the diff eq. kM

Let us consider the center as shown in the figure. The coordinate of the bob of mass m is x1,z1.…

Answered: 1. A pendulum consist of a particle…

Similar questions

choices are stable or uunstable
Assume that you do not know about the kinetic energy or Newton’s Laws of motion. Suppose instead of deriving the Euler-Lagrange equations, we postulate them. We define the basic law of mechanics to be these equations and ask ourselves the question: What is the Lagrangian for a free particle? (This is a particle in empty space with no forces acting on it. Be sure to set up an inertial reference system.) The simplest choice might be to guess it must be proportional to v2,where v→ is the particle velocity in an inertial frame K. Take L = v2. A second inertial frame K′ moves at constant velocity −V0→ with respect to K, so that the transformation law of velocities isv'→ = v→ + V0→. Prove that L' = v'2 is a possible choice for the Lagrangian in the frame K′. Explain how this proves that all inertial frames are equivalent. With this approach we prove the equivalence of inertial frames from the form of the Lagrangian, instead of postulating this equivalence at the start, which is usual way…
The Newton–Raphson method for finding the stationary point(s) Rsp of a potential energy surface V is based on a Taylor-series expansion around a guess, R0. Similarly, the velocity Verlet algorithm for integrating molecular dynamics trajectories [xt, vt] is based on a Taylor-series expansion around the initial conditions, [x0, v0]. Both of these methods rely strictly on local information about the system. How many derivatives do we need to compute in order to apply them?
Lagrangian Dynamics Ep = 0 A pendulum of length / and mass m is mounted on a block of mass M. The block can move freely without friction on a horizontal surface as shown in the adjacent figure H. 1. Find the velocity of mass m, w.r.t the origin O 2. Write the Lagrangian of the system 3. Derive the Euler Lagrange equations
5. Consider the two state system with basis |+) which diagonalizes the Pauli matrix 03. Generally the state of the system at time t can be written as |W(t)) = c+(t)|+) + c_(t)|-). (i) For the Hamiltonian of the system, first take H = functions c+(t) given the initial condition that at time t = 0 Eo03. Solve for the coefficient |W(0)) = |-).
A dart gun contains a spring of spring constant k = 16 N/m which is used to shoot a dart of mass m. The dart leaves the gun at a speed of v = 3.1 m/s after the spring is compressed 1 cm. what is the darts speed when it hits the floor (vf in m/s) if it is fired horizontally at a height of h = 2 m what angle in degree does the darts final velocity make with the horizontal?