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 m

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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 M 1 (61.64 KiB) Viewed 31 times

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 F. = -2m70s. Using N2, write down a differential equation describing the displacement of the mass from its equilibrium position, z(t), then determine the solution for r(t) in the most general form possible by assuming a trial solution x = 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 is very small so you can ignore additive terms of order or higher, prove that the total energy in the system goes down with time as E(t) = [5] -21 [5] – Kx where x=(:) [5] 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 x (t) and r(t). Express these coupled equations in matrix form as in fx M dra i.e. determine the matrices M and K for this system. d. Use the determinant method to solve the matrix equation from part (©), 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 r along the incline, and solve it to determine a general form for 2(t). [5] [5]