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Modeling the Motion of a SpringConsider a weight attached to a springthat is suspended froin a horizontal baras illustrated in the figure. When theobject comes to rest we say it is at"equilibrium" which is labeled 0 on thevertical number line. If you give theweight a push, either up or down, it willstart to move and the motion can bemodeled by sine and cosine functions.The "stiffness" of the spring and themass of the object affect how far the 0object moves from the equilibriumposition. The initial velocity and initialposition also affect the motion of the 2spring. (We don't always start at theequilibrium position.) 4

=

If we neglect any damping forces (airresistance etc.) then the motion of thespring can be modeled by

x(t) = Vosin(wt)+xocos(wt)ca

6

where x(t) is the position of the object along the number line at time t.The otherquantities are constants: OJ is a constant that depends on the stiffness of the spring andthe mass of the weight, vois the initial velocity, and Xoisthe initial position of the object.

Model the motion of a weight on a spring:

Suppose a weight is set in motion from a position 3 centimeters below the equilibriumposition and with a downward velocity of 4 centimeters per second. (please note that thevertical number line used for position is "upside down". This is a convention fromphysics and it means that positions below equilibrium actually correspond to a positivevalue.) Assume that the spring stiffuess and mass of the weight mean that ca = 2 for thissystem.

Part I1) Write the function x(t) that gives the position of the weight as a function of time tinseconds. (Your function should consist of a sine term and a cosine tenn.)

2) Graph the separate sine and cosine components of your function from (1) on the same

set of axes. That is graph ~ = Vosin(mt) and X2 =Xocos(l1Jt)onthe set of axes below.. m .(Sketch these graphs by hand and show two full cycles.)

-5

3) Use a graphing calculator (or online graphing utility) to graph the entire function frompart (1). Use the window settings indicated below. Sketch what you see on yourcalculator display. ~o.y. '" s. ~\ '" ,..,,'1\ -=. - '3, to I

- , \.

xmin=Oxmax=2"xsc1= fymin=-5ymax=5yscl = 1 , ~--~~--~--~~~~--~~~~--~~~~--~-,

-54) Write an equation for your calculator graph in the form x(t) = A cos[ B(t - C)]. (Usethe trace or maximum feature of your graphing utility to help you find values for A, B,and C. I expect to ~ decimal approxim~ese...wlues.L-_----........,

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5) How are the graphs from part 2) related to the graph in part 3)? Are the values forperiod and amplitude the same or different? Why do you think we see these results?Please write out your explanation using complete sentences.~e- C>.>rc.rh lY\ f""'t,- 5 I'" o. rcs."ll DF- 'n....e ("",f:"hc,f,'oll o {- H-L- f)/j)o~k<; tV\. trAIt 2.~ p'~ri()J.'i. ~~ vv-; 'S ,,,../-,'1;c.; (514 (,.')h.{' ~("PL.s.2 V !-3.\5 fur /l-.e ~rp j.. ,;n fr..d 1j["e+~ 1"'4 5f¥h..s..-.-'(iI\ fv+ <- g f4.. "..n;oi.- IV) f"..r( '3. ~ '0/-;0 kA-._J:.e.~ e;rC.

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6) Prove ~ti:foJ~wing is an identity (A is a positive constant) by filling in the blanksbelow.

Asin~sin(mt)+ Acos;cos(mt) = Acos(mt-;)

=A [eos( mt) cos; +sin(mt) sin~] Commutative Property ofMultiplication

A sin ~ sinemt)+ A cos; cos(mt)

=A cos(mt,-~) Difference identity forcosine

Part ill:Rewrite an expression of the form

<Dr ~15]sinemt) +f5] cos( mt) = ~sin( mt) +IAcos fSlcos( mt) in terms of a cosine function:(jj- =""

A cos( mt -~) . Use the following definitions and your result from part II: (\ II

/j'(f' C VlSUo.Jl ~\Nl.u<D c1=AsinfS~sin¢=5. ) C t!)'~~'" Ybo.,st. ~1J..,~

A tan~=_l o- C.C c2 I C, t.a fV\~@ c2 = A cos ~ ~ cos (J =....l:.. " 'r/. \) .

~~~ __ ~A~=~..... ~ YJ pOSS\Dtj.

IA~~<;'+~I ~ ~o.l"i~ .A, is the amplitude of the cosine function and ;, "phi", is called the phase angle and ismeasured in radians;

,Example: Write 4 sin 3t + 2 cos 3t in terms of a cosine function.

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J.. 1:0$\ 91 ~ ~;: ~

>i=- -tM' (1..) ~ \. \ d-It

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··hi",:,-1:: + 'l.(0 S3t 'j:; ';;US co5 (~-I:- \ .\:l.)

~ a(5 c.os\:(t .;0 .~:t)1

7) Rewrite your function from part 1) in terms of acosine function: A cos( ox - ¢). Showyour work. I '&):: --2-~~~-tJJY-+--'-s-~o~tiL'') , _

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