The initial value problem, IVP, has the form In this problem ut) is the position

Question

The initial value problem, IVP, has the form In this problem ut) is the position as a function of time t, the cgefficients a, b, and c are contants, the function gtt) is the external forcing function, the positive reference direction is down. Determine the IVP for the motion of an unforced mechanical mass-spring system with no damping that has a mass of 10 ibs that stretches the spring 6 inches and then is set into motion by displacing it an additional 8 inches upward and providing an initial velocity of 6 inches/sec downward. The numerical values of the coefficilents and initial conditions should use unit in lbs, feet, and seconds. Do not simplify or give algebraically equivalent expressions for the coefficients; express the coefficients in decimal form accurate to at least four places. (Use 32t/sec as the acceleration of gravity. In this problem the downward direction is positive.) Coeffficient of u'tt) 0.012 Coe/fficient of utt)- Coeffficient of ut) 294 Value of u(0) Value of uto)

Explanation / Answer

Sol:

We first need to set up the IVP for the problem. We need to find m and k.

m = w /g = 10/32 = 5/16 = 0.3125

Now, lets find k. We can use the fact that mg = kL to find k.

first we convert 6 inches into foot i,e 0.5 foot

now

k = mg/ L = 10 / 0.5 = 20

We can now set up the IVP.

0.3125 u" + 20 u =0

u(0)= -8  

u ' (0) = 6

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