A simple pendulum consists of mass m suspended by a massless string of length f as shown in the figure to the right. (a) Determine the Lagrangian for this pendulum in terms of its length e and angle 0. (b) Determine Lagrange's equations of motion for this pendulum. m f ite lennth and

damped harmanic oscillator, haS damping constant a = 2 We, that is acted upon by a driving force F = Fo sin wt, The system Starts from rest With an and initial displacement of Xo lie,xLO) = Xo 10)=0), Find the equation of motion and its corres panding salution xlt), determine all of the coefficients le.g., Ai,A2, B,, Bzretc) %3D Be,Bz,et c)

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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