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A.1 Spring Oscillators a. A mass m is attached via a spring with spring constant k to a wall, with the mass allowed to move along a horizontal surface. The surface provides a resistive force proportional to the velocity Fx = -2myuz. Using N2, write down a differential equation describing the displacement of the mass from its equilibrium position, x(t), then determine the solution for c(t) in the most general form possible by assuming a trial solution x = elt. b. Rewrite the solution from part (a) in amplitude-phase form (you don't need to derive this, just state it). Under the assumption that y is very small so you can ignore additive terms of order y or higher, prove that the total energy in the system goes down with time as 1 -2yt E(t) = zkaže c. For the remaining parts, assume the surface is frictionless. A second mass m is attached to the first mass with another spring also having a spring constant k. The second mass is not connected to anything else. Determine a system of coupled differential equations describing the displacements of the two masses xi(t) and x2(t). Express these coupled equations in matrix form as in -d²x M dt2 = -Kx where x= (2) 21 C2 i.e. determine the matrices M and K for this system. d. Use the determinant method to solve the matrix equation from part (c), and deter- mine the frequencies of the normal modes of this system. e. A single mass m and spring with spring constant k is placed on a frictionless incline with an angle • with respect to the horizontal. The spring is affixed to the top of the incline, and the mass hangs from it, free to move along the incline. Draw a force diagram showing all forces on the mass, and decompose the forces into a component along the incline and normal to it. Write down a differential equation for the displacement of the mass x along the incline, and solve it to determine a general form for e(t).