The Lagrangian equation of motion for a system of spring and a mass m connected by a pulley(disk) of radius a as shown in the figure is given by: (a) ë- 28 , 2g , 2ke = 0 За Зт (b) ӧ+ За 2g , 2ko = 0 3m (с) ө— За 2g , ke = 0 2g, ko (d) ô + За m 0 =

In the figure, two satellites, A and B, both of mass m = 36.1 kg, move in the same circular orbit of radius r = 7240 km around Earth (mass ME = 5.98×1024 kg) but in opposite senses of rotation and therefore on a collision course. (a) Find the total mechanical energy Е+ ЕÅ of the two satellites + Earth system before the collision. (b) If the collision is completely inelastic so that the wreckage remains as one piece of tangled material (mass = 2m), find the total mechanical energy immediately after the collision. (c) Just after the collision, is the wreckage falling directly toward Earth's center or orbiting around Earth? (a) Number Units (b) Number (c) Units A B Earth

A cube of a mass m=0.37 kg is set against a spring with a spring constant of k1=656 N/m which has been compressed by a distance of 0.1 m. Some distance in front of it, along a frictionless surface, is another spring with a spring constant of k2=181N/m. The cube is not connected to the first spring and may slide freely. I found (a) and (b). I just need help with (c). a). How far d2, in meters, will the second spring compress when thee cube runs into it? 0.19 m. (answer) b). How fast v, in meters per second, will the cube be moving when it strikes the second spring? 4.21 m/s (answer) c). Now assume friction is present on the surface in between the ends of the springs at their equilibrium lengths, and the coefficient of kinetic friction is uk=0.5. If the distance between the springs is x=1m, how far d2, in meters, will the second spring now compress?

Consider the “Foucault pendulum”, as shown below. Foucault set up his 1851 spherical pendulum (of mass m and length L) experiment in the Pantheon dome of Paris, showing that the plane of oscillation rotates and takes about 1.3 days to fully revolve around. This demonstrated the extent to which Earth’s surface is not an inertial reference frame (e.g., role of the Coriolis force). Your task here is to determine (but not solve) the equations of motion.

Consider the nonconservative mass-spring system governed by ï +2 +26x = 0, x(0) = 1, ż(0) = 4 (a) Find the solution z(t) and its derivative i(t), and evaluate x(7/5) and (7/5). (b) Calculate the total energy E (t) of the system when t = π/5. (c) Calculate the energy loss in the system due to friction in the time interval from t = 0 to t = π/5. no.

A ball is held at rest at some height above a horizontal surface.  Once the ball is released it falls under gravity, hits the surface at time T1 , and starts bouncing vertically up and down. Suppose that with each bounce the ball loses a fixed fraction p (with 1>p>0) of its energy. This loss could be due to a number of reasons (inelasticity, drag, etc) that are left unspecified. How many times will the ball bounce before coming to rest? Provide a detailed explanation of your reasoning, not simply a one-line answer. How long will it take for the ball to come to rest (if at all), i.e., what is the stopping time Tstop? Give your answer for Tstop as a single formula that contains only two independent variables, namely p and the time T1 .