
#37
Oct1208, 10:40 AM

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P: 5,004

x''1000x= (36/m)cos(pt) 10=18cos(pt)10 =>p^2Ae^ipt Bp^2e^ipt1000(Ae^ipt B^2e^ipt)=18cos(pt)10 But what is cos(pt) in terms of complex exponentials? 



#38
Oct1208, 10:57 AM

P: 569

e^{i[tex]\theta[/tex]}= cos([tex]\theta[/tex])+sin([tex]\theta[/tex]) therefore 18*e^{ipt}10= 18*(cos(pt) + sin (pt))10 



#39
Oct1208, 11:02 AM

HW Helper
P: 5,004

But you don't have 18 e^ipt! You have 18 cos(pt).
What is cos(pt) in term of complex exponentials? Hint: look under the section "Relationship to Trignometry" here 



#40
Oct1208, 11:40 AM

P: 569

sorry if you are becoming frustrated. I have a copy of Gregory Douglass's Classical mechanics books and there is an example like this in that book on p. 10 cos(t) and they say the comple counter part is 10e^it 18 cos(pt)= 18e^ipt  10(e^0) 



#41
Oct1208, 12:13 PM

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P: 5,004

cos(pt)≠e^ipt
Euler's formula gives cos(pt)=Real[e^ipt]=(e^ipt+e^ipt)/2 (Real[z] is the real part of the complex number z) so, 18cos(pt)10=9e^ipt+9e^ipt10 =>p^2Ae^ipt Bp^2e^ipt+1000(Ae^ipt +Be^ipt+C)=9e^ipt+9e^ipt10 Compare the coefficients in front of each of the e^ipt,e^ipt, and constant terms...what must A,B and C be (in terms of p)? 



#42
Oct1208, 01:40 PM

P: 569

B(1000p^2)=9 ==>B=9/(1000p^2) C=10/1000? 



#43
Oct1208, 02:02 PM

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P: 5,004

Close, there were some typos in my last equation; you should get A=9/(1000p^2), your B and C are correct though.
Now, since A and B are equal you have: x(t)=A(e^ipt+e^ipt)+C=2Acos(pt)+C Now, if the maximum extension of the spring is 4cm, what must the Value of A be? 



#44
Oct1208, 02:20 PM

P: 569





#45
Oct1208, 02:25 PM

HW Helper
P: 5,004

Yes, since x(t)=Acos(pt)+C, it should be clear that the angular frequency of oscillation is p. So you want to find p.
You need to be careful of your units though; 4cm =0.04m so you should have: 0.04=9/(1000p^2)*cos(pt)1/100 since the rest of the quantities in the equation are in meters. 



#46
Oct1208, 02:29 PM

P: 569





#47
Oct1308, 06:25 PM

P: 569

Sorry to bumped this thread again, even though its been two days since its been active.
for my final solution I get c=36/(1000p^2) and c=.04 ==> p1=10 and p2=10; I'm not sure what my text means when it says 'spring is safe if p=<20 rad/s and p>=40. rad/s 


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