Question

(10 pts) 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) = 170 cos(10t). 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: y'+100y=17cos(10t) help (equations) Initial conditions: y(0) = 0 and y'(0) = 0 help (numbers) C. Solve the initial value problem for y(t). y(t) = U help (formulas) d. Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 <t<. If there is no such maximum, enter NONE. maximum excursion = 170/10(sint) meters help (numbers)

Answer 1

mass is

kg

m = 10

.

spring stretches 9.8 cm or 0.098 meter

so x=0.098

from Hooke's law,

F=kr

mg = kr

10(9.8) = k(0.098)

98 = k(0.098)

N/m

c = 1000

.

.

there is no damping so damping constant is c=0

.

force is

f(t) = 170 cos(100)

.

DE is given by

my+cy + ky = f(t)

104 + Oy + 1000y = 170 cos(100)

104 + 1000y = 170 cos(100)

y + 100y = 17 cos (106)

.

.

initially, spring is in steady-state so ${\color{Red} y(0)=0}$

and there is no initial velocity so ${\color{Red} y'(0)=0}$

.

for homogeneous system

find roots

$x = \pm10i$

for 2 complex roots, the complementary solution is

${\color{blue} y_c=c_1\cos \left(10t\right)+c_2\sin \left(10t\right)}$

.

.

here we have

here $g\left(t\right)=17\cos \left(10t\right)$

which is already a root of given DE so we need to multiply "t" in particular solution

so assume that particular solution is

$y=a_0t\sin \left(10t\right)+a_1t\cos \left(10t\right)$

take first derivative

$y'= a_0\left(\sin \left(10t\right)+10t\cos \left(10t\right)\right)+a_1\left(\cos \left(10t\right)-10t\sin \left(10t\right)\right)$

take second derivative

$y''= a_0\left(20\cos \left(10t\right)-100t\sin \left(10t\right)\right)+a_1\left(-20\sin \left(10t\right)-100t\cos \left(10t\right)\right)$

.

.

put all values in given DE

.

$a_0\left(20\cos \left(10t\right)-100t\sin \left(10t\right)\right)+a_1\left(-20\sin \left(10t\right)-100t\cos \left(10t\right)\right)+100(a_0t\sin \left(10t\right)+a_1t\cos \left(10t\right))=17\cos \left(10t\right)$

.

$20a_0\cos \:\left(10t\right)-100a_0t\sin \:\left(10t\right)-20a_1\sin \:\left(10t\right)-100a_1t\cos \:\left(10t\right)+100a_0t\sin \:\left(10t\right)+100a_1t\cos \:\left(10t\right)=17\cos \:\left(10t\right)$

.

$20a_0\cos \left(10t\right)-20a_1\sin \left(10t\right)=17\cos \left(10t\right)$

compare coefficient both sides.

$0=-20a_1........................{\color{Blue} a_1=0}$.

$17=20a_0........................{\color{Blue} a_0=\frac{17}{20}}$

put both constant in a particular solution

$y=\frac{17}{20}t\sin \left(10t\right)+0\cdot \:t\cos \left(10t\right)$

$y=\frac{17t\sin \left(10t\right)}{20}$

${\color{blue} y_p=\frac{17t\sin \left(10t\right)}{20}}$

..

general solution is

$y=y_c+y_p$

$y=c_1\cos \left(10t\right)+c_2\sin \left(10t\right)+\frac{17t\sin \left(10t\right)}{20}$....................(1)

.

here y(0)=0

17. 0 .sin (100) 0 = c cos (100) + C2 sin (100) + + 20

$0=c_1\cos \left(0\right)+c_2\sin \left(0\right)+0$

$0=c_1+0$

....................put it back in equation 1

C1 = 0

.

$y=0\cdot \cos \left(10t\right)+c_2\sin \left(10t\right)+\frac{17t\sin \left(10t\right)}{20}$

$y=\frac{17t\sin \left(10t\right)}{20}+c_2\sin \left(10t\right)$.........................(2)

take derivative

$y'=\frac{17}{20}\left(\sin \left(10t\right)+10t\cos \left(10t\right)\right)+c_2\cos \left(10t\right)\cdot \:10$

here y'(0)=0

$0=\frac{17}{20}\left(\sin \left(10\cdot \:0\right)+10\cdot \:0\cdot \cos \left(10\cdot \:0\right)\right)+c_2\cos \left(10\cdot \:0\right)\cdot \:10$

$0=\frac{17}{20}\left(\sin \left(0\right)+0\right)+c_2\cos \:\left(0\right)\cdot 10$

(0+0) + 02.10

$0=0+10c_2$

$c_2=0$...................put it back in equation 2

.

$y=\frac{17t\sin \left(10t\right)}{20}+0\cdot \sin \left(10t\right)$

${\color{Red} y(t)=\frac{17t\sin \left(10t\right)}{20} }$

.

.

.

from the graph, there is no maximum excursion

so type NONE

.