Problem number ( 9) Your suppose to ket k = (8/9) please show your work 6. Solve

Question

Problem number (9)

Your suppose to ket k = (8/9)

please show your work

6. Solve the initial value problem y 2with y(0)1 and (O)sng the Method of Undetermined Coeficents (Axt) X 7. Find the general solution of the equation y" 4y 3 csc(2r) for 2 using Variation of Parameters Mechanical Vibrations 8. Co nsider a mass-spring system with mass m 3 kg, spring constant and damping constant c . There is no external force applied to the system. 12 (a) Set up a differential equtaion to model the motion of this system, and find the general solution. (b) Is the system underdamped, overdamped, or critically damped? (c) If the system is not critically damped, find a value for c that makes the system critically damped. k Newtons per meter and no damping. Suppose the system is at rest with no external force, and at time t 0 the mass is kicked and starts traveling in the positive direction at 2 meters per second. 9. A mass of 2 kilograms is on a spring with spring constant How large does k have to be so that the mass does not go further than 3 meters from the rest position?

Explanation / Answer

Sol:

Take the zero point of the potential energy to be the "rest" position of the mass. Let this be the position, x = 0. I will also assume that the spring is oriented horizontally, so we don't have to figure in the effects of gravity.

The mass initially has zero potential energy, but as a velocity of 2m/sec, so its kinetic energy is:

KE(t = 0) = (M/2)*v^2 = (2kg/2)*(2m/s)^2 = 4 J

At the extrema of its travel, the mass comes to rest for an instant, so its kinetic energy is zero, but the mass-spring system has a potential energy of:

U = (k/2)*x^2

where x is the displacement from the rest position (x = 0).

At the maximum displacements, all the kinetic energy initially imparted to the mass has been converted to potential energy by compressing or extending the spring, so at those points:

4 J = (k/2)*x^2

We want to know how large k must be to keep |x| < 3 m, or to keep x^2 < 9 m^2

x^2 = (4J)*2/k

9m^2 > (4J)*2/k

k > (8 J)/(9 m^2)

k > (8/9) kg/s^2 ~= 0.889 kg/s^2

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