Obtain the Lagrange equations of motion for a spherical pendulum, i.e., a mass point suspended by a rigid weightless rod.

Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.

A spring mass system consists of a sping with spring constant k and an attached block of mass m is submerged in a liquid that produces a damping force Fr.  m=2kg Fr= 18 times the instaneous velocity of the center of mass of the block  k=36 n/m If the mass is initially released from rest 1 meter below equilibrium position  a. Give a 2nd degree equation that describe the motion of the center of mass of the attached block  b. Solve the equation in part a

A horizontal spring mass system oscillates on a frictionless plane.  At time t=0, it is moving left at position x=9 cm.  It has velocity v=0, at positions x=0 cm and 12 cm, and completes one full cycle in 2 seconds.  Write the position and velocity kinematic equations for this oscillating system, including the phase constant.

The system in the figure below is in equilibrium, and its free body diagram drawn on the right. The distance, d is 1.14 m and each of the identical spring's relaxed length is l0 = 0.57 m. The mass, m of 0.86 kg brings the point P down to a height h = 15 cm. The mass of the springs are negligible. Calculate the following quantities: (a) The angle ? (b) The force exerted on P by the right spring (c) The force exerted on P by the left spring (d) The total spring length (e) The stretch length (f) The stiffness constant of the springs

Develop a Lagrangian for the double pendulum. You may need to make some assumptions to simplify the problem. You may also need to introduce some new variables to make the problem work. Make sure that is explained.

A block of mass m = 0.52 kg attached to a spring with force constant 143 N/m is free to move on a frictionless, horizontal surface as in the figure below. The block is released from rest after the spring is stretched a distance A = 0.13 m. Refer to attached Image. (a) At that instant, find the force on the block. (b) At that instant, find its acceleration.