A 10 kilogram object suspended from the end of a vertically hanging spring stret

Question

A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 20cos(l0t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds. a. Determine the spring constant k. k = 1000 Newtons/meter b. Formulate the initial value problem for y(t), where y{t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y', y",t. Differential equation: i0y"+1000y-20cos(10t) Initial conditions:;y(0) = and y (0) = 0 c. Solve the initial value problem for y(t). y(t) = d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0

Explanation / Answer

mg = kx

10(9.8) = k(9.8)(1/100)

=> k = 1000

Since object is released from equilibirium , initially at rest

=> y(0) = 0 and y'(0) = 0

General Equation for a forced Simple Harmonic Oscillator

=> mx'' + kx = F(t)

=> 10y'' + 1000y = 20cos(10t)

=> y'' + 100y = 2cos(10t)

Solutions of homogenous equation :

y'' + 100y = 0

=> y = c1cos(10t) + c2sin(10t)

=> y' = -10c1sin(10t) + 10c2cos(10t)

Since y(0) = 0 => c1 = 0

and y'(0) = 0 => c2 = 0

For particular solution , let y(t) = (A + Bt) cos(10t) + (C + Dt) sin(10t)

y' = - 10(A + Bt) sin(10t) + 10(C + Dt) cos(10t) + Bcos(10t) + Dsin(10t) = (10C + B + 10Dt)cos(10t) + (-10A -10Bt + D)sin(10t)

y'' = -10(10C + B + 10Dt)sin(10t) + 10(-10A -10Bt + D)cos(10t) + 10Dcos(10t) -10Bsin(10t)

y'' = cos(10t)( -100A -100Bt + 20D ) + sin(10t)( -100C -20B - 100Dt )

100y = (100A + 100Bt) cos(10t) + (100C + 100Dt) sin(10t)

y'' + 100y = 20Dcos(10t) -20Bsin(10t) = 2cos(10t)

=> A = C = B = 0 and D = 1/10

=> y(t) = (t/10)sin(10t)

Since the function increasing from 0 to infinity.Hence,there is no maximum excursion

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