Can someone explain to me what my professor did?

In summary: So, the sum of the two can also be written as:##x(t) = c_1 A+c_2 \cos (\omega t+\phi)##In summary, the homework equation is:x(t) = c_1 cos(ωt) + c_2 sin(ωt) where c1 and c2 are constants. The sin and cos terms can be amalgamated into a single cos (or sin) term with a constant phase shift.
  • #1
Warlic
32
0

Homework Statement


upload_2015-11-6_17-35-30.png
[/B]

Two things I don't understand; how did he get that omega is sqrt(k/m-(b/2m)^2)
And second; why is it that x(t) = e^(-bt/2m)* cos (omega*t+phi)
shouldnt it rather be; x(t) = c1*cos(ωt) + c2*sin(ωt)

Homework Equations

The Attempt at a Solution

 
Physics news on Phys.org
  • #2
I figured out why omega is what it is, but still don't understand how he got the equation for x(t)
 
  • #3
I think he's assuming that the system is underdamped, so will be a damped sinusoid. The general solution can be molded into that shape:

##x(t) = c_1 cos(ωt) + c_2 sin(ωt)##
##~~~= \sqrt{c_1^2 + c_2^2}\left( \frac{c_1}{\sqrt{c_1^2 + c_2^2}} cos(ωt) + \frac{c_2}{\sqrt{c_1^2 + c_2^2}} sin(ωt) \right) ##
##~~~= \sqrt{c_1^2 + c_2^2}\left(cos(\phi) cos(ωt) + sin(\phi) sin(ωt) \right) ##
##~~~= \sqrt{c_1^2 + c_2^2} cos(ωt - \phi)## where: ##~~~~\phi = tan^{-1}\left( \frac{c_2}{c_1} \right)##

You can fudge the sign of the angle ##\phi## by negating it and interpreting it appropriately.
 
  • Like
Likes Warlic
  • #4
Aren't c1 and c2 constants? If so, I see a sinusoid, not a damped sinusoid.
 
  • Like
Likes Warlic
  • #5
Mister T said:
Aren't c1 and c2 constants? If so, I see a sinusoid, not a damped sinusoid.
Ah, right. I left out the damping term in the general solution o:) So:
##x(t) = e^{-\alpha t}(c_1 cos(ωt) + c_2 sin(ωt))##
Go from there. The roots of the auxiliary equation will be complex conjugates of the form ##\alpha ± \omega##, where ##\omega## can be further broken down as ##\omega = \sqrt{\alpha^2 - \omega_o^2}##
 
  • Like
Likes Warlic
  • #6
gneill said:
Ah, right. I left out the damping term in the general solution o:) So:
##x(t) = e^{-\alpha t}(c_1 cos(ωt) + c_2 sin(ωt))##
Go from there. The roots of the auxiliary equation will be complex conjugates of the form ##\alpha ± \omega##, where ##\omega## can be further broken down as ##\omega = \sqrt{\alpha^2 - \omega_o^2}##
This is exactly the point where I don't know where to go from :P. Where does the c2sin(ωt) part go?
 
  • #7
Warlic said:
This is exactly the point where I don't know where to go from :P. Where does the c2sin(ωt) part go?
See post #3. The sin and cos terms can be amalgamated into a single cos (or sin) term with a constant phase shift.
 
  • Like
Likes Warlic
  • #8
Look at the undamped case:

##x(t)=c_1 \cos (\omega t)+c_2 \sin (\omega t)##

and

##x(t)=A \cos (\omega t+ \phi)##

are equivalent. Note that each contains two constants of integration.
 
  • Like
Likes Warlic

1. What exactly did my professor do?

Your professor likely taught a lesson or gave a lecture on a particular subject. It is also possible that they gave an assignment or administered a quiz or test.

2. Why did my professor do what they did?

Your professor's actions were most likely in line with the course curriculum and objectives. They may have also been trying to help you learn and understand the material better.

3. How does what my professor did relate to the course material?

Your professor's actions were likely directly related to the course material and topics being covered. They may have provided examples or explanations to help you better understand the material.

4. Can someone explain my professor's teaching style?

Your professor may have a particular teaching style that they use in their lectures or lessons. This could include using visual aids, group activities, or other methods to engage students and help them learn.

5. How can I better understand what my professor did?

To better understand what your professor did, you can review your notes, ask questions during class or office hours, or seek clarification from classmates. You can also do additional research on the topic to deepen your understanding.

Similar threads

  • Introductory Physics Homework Help
Replies
17
Views
224
Replies
20
Views
788
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
868
  • Introductory Physics Homework Help
Replies
9
Views
608
  • Introductory Physics Homework Help
2
Replies
35
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
608
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
1K
Back
Top