Question

A mass of m kilograams (kg) is mounted on top of a vertical spring. The spring is L metres long when disengaged and the end not attached to the mass is fixced to the ground. The mass moves vertically up and down, acted on by gravity, the restoring force T of the spring and the damping force R due to friction: see the diagram below The gravitational force is mg dowswards, where g- 9.8m is acceleration due to gravity, measured in metres per second squared. Let k denote the spring constant, measured in Newtons per metre (N/m) and let B be the damping constant, measured in Newton seconds per metre (N x s)/m). The spring obeys Hooke's Law and is compressed by a length s metres when the restoring force balances gravity. In addition, the damping force R, with magsitude the constant B multiplied the velocity of the mass, acts on the mass in the direction opposite to its motion. We assume the dowaward direction is the positive direction and let y(t) be the displacement in metres of the mass from its equilibrium position at a time t, measured in seconds ise no ye mass ounol (a) Using Newton's Second Law, write down an equation relating m, the acceleration of the mass and the three forces, gravity, 7 and R, acting on the mass. Hence deduce that y satisfies the equation of motion: (b) We take measurements and find that m-1, k-5 and B-4 (0) Determine if the system is under-damped, critically damped or over-damped. Find the general solution to the equation of motion. (") Initially the mass moves downward with speed 1(m/s) from the equilibrium position. Solve the initial value problem to find the function t). (ili) Continuing from Part (il), determine whether the mass is ever at the equilibeium position again. If it is, find the smallest positive value of time, to for which y(te)-0 Justify your ans wers (c)Assume still that m-1 and k 5, but now the device is moved into vacuum so that there is no damping; that is B-0 () Determine the general solution to the equation of motion wben there is no external force applied. (ii) Assume now that a dowaward force fit)-A cot-)) is applied to the mass. For what values of A, w and φ does resonance orur in the system? Justify your answers. Page 2 of 2

Answer 1

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2: at tO -210)

(-2F)L 2-1 2-T 上一上 To reach 1stもme at maan position Б: 2 - = 2-14 see

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