A mass that weighs 8 lb stretches a spring 6 in. The system isacted on by an external force of 8 sin 8t lb. If the mass is pulleddown 3 in and then released, determine the position of the mass atany time. Determine the first four times at which the velocity ofthe mass is zero.

t= 1/8, pi/8, 2pi/8,3pi/8

Step-by-step explanation:

Given

m=(8/32) lb s^2/ft

K=8/(6/12)=16 lb/ft

Use the following equation and plug in values

mu''+ku=f(t)

1/4u''+16u=8sin8t

u''+64u=32sin8t

This equation corresponds to the following homogeneous equation

u''+64u=0

r=+/-8i

uc(t)=c1cos8t+c2sin8t

Now find the particular solution

u(t)=Atcos8t+Btsin8t

u'(t)=-8Atsin8t+Acos8t+B8tcos8t+Bsin8t

u''(t)=-8tAsin8t-64Atcos8t-8Asin8t+B8cos8t-64Btsin8t+8Bcos8t

Substitute these values into the original equation and solve for Aand B

A=-2 B=0

the particular solution is u(t)=-2tcos8t

the general solution is u=u1(t)+u(t)

u=c1cos8t+c2sin8t-2tcos8t

Use the initial conditions to solve for c1 andc2

c1+0=(1/4)   8c2-2=0

c1=(-1/4) c2=(1/4)

u=(1/4)[cos8t+sin8t-8tcos8t]

To solve the next step differentiate u

u'=-2sin8t+2cos8t-2cos8t+16tsin8t

= -2sin8t+16sin8t

= 2sin8t(8t-1)

Velocity=2sin8t(8t-1)

Set this equation equal to zero to solve for zero velocity

8t-1=0 t=1/8

t= 1/8, pi/8, 2pi/8,3pi/8

the answer is n

answer: the value should be used for i in the formula of calculation the monthly payment , the interest rate per period is 0.85.

step-by-step explanation:

the given rate of interest i= 85%

in calculating the monthly payment for a loan with an interest rate compounded monthly, the value of the interest rate per period we use in the formula must be in decimal form.

to convert it in decimal, we divide it by 100, we get

hence, the value should be used for i in the formula of calculation the monthly payment , the interest rate per period is 0.85.