An 8-kg mass is attached to a spring hanging from the ceiling and allowed to com

Question

An 8-kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. Assume that the spring constant is 40 N/m and the damping constant is 4 N-sec/m. At time t=0, an external force of F(t)=4cos(2t+pi/4) is applied to the system. Formulate the initial value problem describing the motion of the mass and determine the amplitude and period of the steady-state solution. Let y(t) to denote the displacement, in meters, of the mass from its equilibrium position. Set up a differential equation that describes this system. (give your answer in terms of y'', y', y). Also show the amplitude of the steady-state solution and the period of the steady-state solution.

Explanation / Answer

m*y''+k*y=F(t)

m*y''+k*y=4cos(2t+Pi/4)

see we can write accleration as y" because its double derivative of y when y is changing

i.e

dy/dt= v= y'

dv/dt= a= y''

k*y is the spring force

and 4*v is the damping force and it is against the motion so it comes with a negative sign

so we say:

m*a +k*y- 4*y'= Total force w.r.t time=4cos(2t+Pi/4)

solving we will get two solutions

1) complementary function

2) particular solution

1)

y= Acos(?t)+Bsin(?t)

2)

y=4cos(2t+Pi/4)/?^2

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>?=?k/m=?5

the second solution is steady state solution

a)so time period of steady solution is =2?/?f=2?/2=?

amplitude = 4/?^2=4/5=.8

see now i explained each term used

now please rate me 5 stars.

thank you.

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