professor, before you sleep every night remember that someone , somewhere is watching your lectures and getting inspired. we thank you for your service to science
I think it's neat that the equations for x1 and x2 at 15:35 look strikingly similar to the right/left basis vectors as functions of the up/down basis vectors in quantum mechanics! |r> = (1/sqrt(2))|u> + (1/sqrt(2))|d> and |l> = (1/sqrt(2))|u> - (1/sqrt(2))|d> They only differ by a +/- sign in the middle!! Is it purely coincidental that these are so similar? Love your lectures!
this is the most boring topic ever! (you have to go through all the math, making sure the signs are correct and make sure you didnt make a slip of the pen), but you are able to make the lesson so interesting that none of them matters anymore! I love you dr lewin!!
53:45 I think there was a faster way to find out the C1/C2: Since the two reciprocal numbers equals, we immediately get positive and negative 1. Right?
I'm sorry if you adressed this in other lecture that I didn't recall, but why not having damping on this system means that the normal modes will be either in phase or out of phase (around 3:00)?
+Family Father Damping of "some kind" is always needed to kill the transient solutions so that we end up with the so-called "steady state solutions". I discuss transient and "steady state" solutions in detail in my 8.03 lectures. When we achieve a steady state solution, the frequency of the oscillators is the same as the driving frequency. If the driving frequency is a resonance frequency damping of some kind is also a must - without it, the amplitudes would go to infinity. When we excite a stable steady state resonance frequency adjacent oscillators are in phase or out of phase. I have demonstrated that in several ways. As an example if you have a double pendulum, there will be 2 resonance frequencies. In the lowest frequency the 2 pendulums are in phase in the high frequency they are out of phase.
1)Resonance-a frequency or something that is absolutely lovable by the system ,and system begins to picking up speed. 2) standing wave - if a wave has a limited medium to travel ,like in a string of a certain length ,then this is called a standing wave.. 3)If we apply external force with a certain frequency ,then for a while system will not like that and try to oppose the motion but if you wait ,then you will find that system will behave according to your force
In the 3 cars and 4 spring problem, when using the method you presented should I take the ratios 2 by 2? I mean, C1/C2 and C2/C3 and/or C1/C3? I will try to solve this problem using this method. I appreciate so much your lectures! Thank you very much, Sir, you are really helping me studying for my tests.
Professor, when you place the spring right at the top of the two coupled pendula, I can still see the spring elongate and contract. Does this have any effect on the movement of the two coupled oscillators?
Hi Dr. Lewin I have a question about when you were going through Newton's second law at 13:30 or so. You stated that the tension force equals mg, but a y-direction force balance around the object gives T=mg/cos (theta) where cos(theta) = sqrt(l^2-x^2)/l. Just wondering how you got T to be mg or if I missed something here?
For the EOM of X1 there should be a minus sign between modes because X1 object switches direction btw modes not a plus sign and for X2 there should be a plus sign because X2 object does not switch direction btw modes.This is according to the drawings on the blackboard.Wonderfull lecture.
At 27:00 What will happen if I attach two springs one at midway and other at the bottom if both springs are of same spring constant? Also, what will happen if both springs have dissimilar value of spring constant? Sir plz help
Lectures by Walter Lewin. They will make you ♥ Physics. I just found a video about coupled oscillators by Prof. Wit Busza on your channel. Hope that was the video you were talking about. You lectures are helping me a lot for my preparation. Thank you so much!
58:30 The fractions were confusing. I was sure it's C2/C1=1-sqrt2 and thought, how in the world could that be wrong. Only later I realized that it's correct :D
1:12:40 I dont understand it took the guy 20 hours, it's not so terribly complicated. Setting x1=A1 cosωt etc, and defining ω0^2=k/m, Newton's 2nd law gives x1" = -ω^2 A1 = ω0^2 (-2A1+A2) x2" = -ω^2 A2 = ω0^2 (A1-2A2+A3) x3" = -ω^2 A3 = ω0^2 (A2-2A3) So 2 - ω^2/ω0^2 = A2/A1 = (A3+A1)/A2 = A2/A3 Or seting the denominators equal: A2^2•A3 = (A1+A3)•A1•A3 = A1•A2^2 Case 1: A2=0 now it follows that A3=-A1 and ω^2 = 2ω0^2 Case 2: A20 now it follows that A3=A1 and A2/A1=2A1/A2, so that A2^2= 2A1^2 Two subcases: A2= + sqrt(2)•A1, ω^2 = (2 - sqrt(2)) ω0^2 A2= - sqrt(2)•A1, ω^2 = (2 + sqrt(2)) ω0^2
Hello professor. I have trouble choosing a book to study on "Statistical Mechanics" and preparing my-self for next semester. I know the basic thermodynamics and probability distribution. I need a text book that is not too hard or too simple to start and fill the summer with. I found these books to be help full. "Stat Mechanics by Mehran Kardar", "Stat Mechanics by kerson Huang", "Stat Mechanics by Charles Kittel", Stat Mechanics by Walter Greiner", and "Stat Mechanics by Terrell Hill". I would really appreciate if you could guide me which one to study first or you suggest another book?
Sir, I’ve been working on your problem on 1:00:00 with double pendulum for few days and I always get same and different solution from yours, maybe I am making mistake in writing newtons 2nd law for first object, i write that the force is equal to -mgsin(theta1)+T2sin(theta2-theta1) where T2 is the tension on second string caused by second object. Is there any way that you could make a video solving this problem, i really looked up for the solution but almost all of them use lagrangian while solving and make problem more general with strings of differents leghts and object with different mass.
Lectures by Walter Lewin. They will make you ♥ Physics. Yes but they are a lot more complicated (for a sophmore in high school like I am) than your solutions is, so is there any chance that you could solve this problem in one of your future videos or do you maybe know someone who has already done it your way, it would be really reliefing since I’ve been struggling with this for a few days and i would really like to know how to solve this
Easiest way is to compute the kinetic and potential energy of the system and use the Euler-Lagrange equations: electron6.phys.utk.edu/PhysicsProblems/Mechanics/6-Oscillations/coupled-spring.html. You can also use symmetry and conserved quantities to find the frequencies for many systems with little effort. The example of a linear triatomic molecule is a nice one. The center of mass undergoes uniform translation in one mode so a zero frequency mode exists; and the other two modes are easily visualized and the frequencies found.
Many of the most spectacular discoveries in Physics, at first, had NO connection with Technology. A striking example is Faraday's Law. *It's running our entire economy* When Faraday was interviewed the interviewer questioned the importance of his discovery and he asked: "what is the importance of your discovery?" Faraday answered: "The days will come that you will tax it."
Dear profecor EXCELLENT what I can not understand is how a wave with an open bountry does not dump to the end 9 the string and cames back , it is like hiting a rock to the midle of the ocean and the wave comes back to you strange .( 3 days wandered in my head )
www.acs.psu.edu/drussell/Demos/reflect/reflect.html. Open boundaries are a condition on derivatives while hard boundaries are conditions on displacements.
how can we understand the coupling between modes( more precisely anharmonic modes). how can we correlate it with spring mass system. i work with baw resonators.
Another amazing lecture professor! I have a question : why are we able to express any motion of the coupled osscilators as the superposition of the normal modes? I understand this somewhat intuitively but I don't feel I really know the reason.
In fair chance, the metal ball will try to keep balance as you may see the metal ball those far away from the frequency source has more velocity. One thing right is the enery transfering from potentail to kinetic but since the frequency source is keep feeding more energy into the pendulum. Once the energy from the souce is stop then it will try to becomes equilibrium or you may see the multiples metal balls pendulum that some ball will not moving in some frequency and amplitude. But if you want all balls moving feed more energy let's swing over your head! In not a fair chance, where you are trying to send some energy to a nearest ball, in this case it will moving the nearest ball but if you provide more energy the other balls will keep balancing. That is why we are using long chain pandulum to at least absorb some effect of un-fair chance.
Question. I always get the steady state part of the equation correct. I always get the transient wrong. It's clear from my answers that i got part of it correct but, ultimately the answer for my transient solution is incorrect. I've watched your videos 8.03 lecture 5 and was hoping for an answer. You eventually said you'd not be giving the answer for the undetermined coefficients of the transient. I only read math books for fun so not an enrolled student. If you have any time could you give me a couple of most common mistakes? Maybe from that i can figure out what I'm doing wrong . Thank you!
Sir, quick question, how can we know when the system has an extra omega (omega++), since you did not mention the highest mode on your first example (two balls on pendulum connected with a spring)
2 balls on 2 connected strings have 2 resonance frequencies. n balls on n connected strings have n resonance frequencies. Watch my demo with n=3, very cool!
Lectures by Walter Lewin. They will make you ♥ Physics. At 1:04:18 you introduce the term highest mode but you did not mention that on 03:00 on your first example, my question is, why your first example doesnt have tht third mode (omega++)
Lectures by Walter Lewin. They will make you ♥ Physics. Oh thank you very much, I guessed that was it, I did, all of your demos are amazing, but by far the most thrilling one was the one with disproving the kirchoffs rule (8.02 lect 16) :)
2004? WHOOOAAAAAA I was 3 years old.......then.... And finally coupled oscillators......... Should I watch each video from before? Or this as a standalone?
Screw it....I am going for the full course..... I want to have better understanding of physics...and better Thinking .... I will Start from your first problem sir....I couldn't do the Coupled oscillator one as well....
I tried googling, but all the sollutions use energy equations with complicated matrix solving methods. They do not use Newton's law to write the equation.
Lectures by Walter Lewin. They will make you ♥ Physics. wanted to know if it's possible to solve just using the kind of mathematical tools used in this video. With Newton's laws
@@rupeshknn cis.poly.edu/~mleung/CS4744/f03/ch04/linmol.html. The linear algebra makes your life easier; not harder. Lewin is doing the linear algebra without telling you that's what he's doing. Learn the Lagrangian approach. It's systematic and there is less room for error in getting the equations of motion and normal modes.
Sir what is reason that if there are 2 pendulm coupled together then then there will be two normal modes Or What is the reason that if there are n pendulm coupled together then there will be n normal mode
@Proffesor Lewin I don't suppose you have access to demographic statistics regarding who is watching your lectures? I'm rather curious about who is making use of these, given I'm purely here for entertainment.
Hi Professor, and thanks for the wonderful lectures. Just a quick comment. At minute 22, in the second equation (x2), the trig equivalence should be minus 2sin half the sum/half the difference, right?
@@lecturesbywalterlewin.they9259 Hello Prof. Lewin, may I ask then why in your lecture and French's book there is no negative sign in the equation of x2? Thank you in advance.
professor, before you sleep every night remember that someone , somewhere is watching your lectures and getting inspired.
we thank you for your service to science
Thank you Professor Lewin! Still watching and learning from your videos!!! Hope you are doing well.
You are very welcome
Thank you so much for this lecture. Coupled oscillators are something that's been abit of a struggle and this has helped straighten alot of that out.
You're very welcome!
I wish my ME classes had these demonstrations to show that the math actually works.
I think it's neat that the equations for x1 and x2 at 15:35 look strikingly similar to the right/left basis vectors as functions of the up/down basis vectors in quantum mechanics!
|r> = (1/sqrt(2))|u> + (1/sqrt(2))|d> and |l> = (1/sqrt(2))|u> - (1/sqrt(2))|d>
They only differ by a +/- sign in the middle!! Is it purely coincidental that these are so similar? Love your lectures!
this is the most boring topic ever! (you have to go through all the math, making sure the signs are correct and make sure you didnt make a slip of the pen), but you are able to make the lesson so interesting that none of them matters anymore! I love you dr lewin!!
he's so good at explaining these things, I learned and had much fun and amazed from this inherently rather boring topic. Thanks dr. Lewin.
Your lectures really enlighten me,Sir.
Glad to hear that
Professor, at 22:09, isn't the cos a - cos b equal to MINUS the sin of half the sum times half the difference?
At point 22:06 a minus sign is needed.
I found this lecture better than the one given by Professor Lee
53:45 I think there was a faster way to find out the C1/C2: Since the two reciprocal numbers equals, we immediately get positive and negative 1. Right?
if you find the exact same value that I find, then that's fine with me
Yes, thanks!...
I'm sorry if you adressed this in other lecture that I didn't recall, but why not having damping on this system means that the normal modes will be either in phase or out of phase (around 3:00)?
+Family Father Damping of "some kind" is always needed to kill the transient solutions so that we end up with the so-called "steady state solutions". I discuss transient and "steady state" solutions in detail in my 8.03 lectures. When we achieve a steady state solution, the frequency of the oscillators is the same as the driving frequency. If the driving frequency is a resonance frequency damping of some kind is also a must - without it, the amplitudes would go to infinity. When we excite a stable steady state resonance frequency adjacent oscillators are in phase or out of phase. I have demonstrated that in several ways. As an example if you have a double pendulum, there will be 2 resonance frequencies. In the lowest frequency the 2 pendulums are in phase in the high frequency they are out of phase.
Thank you very much sir, those demonstrations make me feel that it more practical, awesome lectures ❤
Sir, In physics what is the most perfect definitions of the following terms: 1) Resonance 2)Standing wave 3)Force Vibrations
ask google
@@lecturesbywalterlewin.they9259 :)
1)Resonance-a frequency or something that is absolutely lovable by the system ,and system begins to
picking up speed.
2) standing wave - if a wave has a limited medium to travel ,like in a string of a certain length ,then this is called a standing wave..
3)If we apply external force with a certain frequency ,then for a while system will not like that and try to oppose the motion but if you wait ,then you will find that system will behave according to your force
@@lecturesbywalterlewin.they9259 savage
In the 3 cars and 4 spring problem, when using the method you presented should I take the ratios 2 by 2?
I mean, C1/C2 and C2/C3 and/or C1/C3?
I will try to solve this problem using this method.
I appreciate so much your lectures! Thank you very much, Sir, you are really helping me studying for my tests.
Professor, when you place the spring right at the top of the two coupled pendula, I can still see the spring elongate and contract. Does this have any effect on the movement of the two coupled oscillators?
Hi Dr. Lewin I have a question about when you were going through Newton's second law at 13:30 or so. You stated that the tension force equals mg, but a y-direction force balance around the object gives T=mg/cos (theta) where cos(theta) = sqrt(l^2-x^2)/l. Just wondering how you got T to be mg or if I missed something here?
JK I'm thinking you used the small angle approximation.
yesssss small angle approx. cos(alpha) is very very close to 1.000. Take 5 degrees, cos(5)=0.996
Thank bro same problem face by me but i clear doubt by your comment
Very good lecture . Thanks and Regards 🙏🙏🙏🙏🙏
For the EOM of X1 there should be a minus sign between modes because X1 object switches direction btw modes not a plus sign and for X2 there should be a plus sign because X2 object does not switch direction btw modes.This is according to the drawings on the blackboard.Wonderfull lecture.
At 27:00
What will happen if I attach two springs one at midway and other at the bottom if both springs are of same spring constant? Also, what will happen if both springs have dissimilar value of spring constant? Sir plz help
I cover this in my lectures and in my problem solving Help Sessions.
Lectures by Walter Lewin. They will make you ♥ Physics.
I just found a video about coupled oscillators by Prof. Wit Busza on your channel. Hope that was the video you were talking about.
You lectures are helping me a lot for my preparation. Thank you so much!
I cover *free oscillations* of springs in 8.01. In my Help Sessions also. But you may not need those anymore.
Lectures by Walter Lewin. They will make you ♥ Physics.
If it's a physics lecture and that too by Prof. Lewin, I WILL watch it! Thank you sir :)
58:30 The fractions were confusing. I was sure it's C2/C1=1-sqrt2 and thought, how in the world could that be wrong. Only later I realized that it's correct :D
1:12:40
I dont understand it took the guy 20 hours, it's not so terribly complicated.
Setting x1=A1 cosωt etc, and defining ω0^2=k/m, Newton's 2nd law gives
x1" = -ω^2 A1 = ω0^2 (-2A1+A2)
x2" = -ω^2 A2 = ω0^2 (A1-2A2+A3)
x3" = -ω^2 A3 = ω0^2 (A2-2A3)
So 2 - ω^2/ω0^2 = A2/A1 = (A3+A1)/A2 = A2/A3
Or seting the denominators equal: A2^2•A3 = (A1+A3)•A1•A3 = A1•A2^2
Case 1: A2=0 now it follows that A3=-A1 and ω^2 = 2ω0^2
Case 2: A20 now it follows that A3=A1 and A2/A1=2A1/A2, so that A2^2= 2A1^2
Two subcases:
A2= + sqrt(2)•A1, ω^2 = (2 - sqrt(2)) ω0^2
A2= - sqrt(2)•A1, ω^2 = (2 + sqrt(2)) ω0^2
Hello professor. I have trouble choosing a book to study on "Statistical Mechanics" and preparing my-self for next semester. I know the basic thermodynamics and probability distribution. I need a text book that is not too hard or too simple to start and fill the summer with. I found these books to be help full. "Stat Mechanics by Mehran Kardar", "Stat Mechanics by kerson Huang", "Stat Mechanics by Charles Kittel", Stat Mechanics by Walter Greiner", and "Stat Mechanics by Terrell Hill". I would really appreciate if you could guide me which one to study first or you suggest another book?
search the web
Sir, I’ve been working on your problem on 1:00:00 with double pendulum for few days and I always get same and different solution from yours, maybe I am making mistake in writing newtons 2nd law for first object, i write that the force is equal to -mgsin(theta1)+T2sin(theta2-theta1) where T2 is the tension on second string caused by second object. Is there any way that you could make a video solving this problem, i really looked up for the solution but almost all of them use lagrangian while solving and make problem more general with strings of differents leghts and object with different mass.
Lagrangian solutions are fine. Since I did not cover that in this course I did not use Lagrangian.
Lectures by Walter Lewin. They will make you ♥ Physics. Yes but they are a lot more complicated (for a sophmore in high school like I am) than your solutions is, so is there any chance that you could solve this problem in one of your future videos or do you maybe know someone who has already done it your way, it would be really reliefing since I’ve been struggling with this for a few days and i would really like to know how to solve this
Do the cameras in front detect the movement of Prof. Walter Lewin?
Thanks in Advance😊
well spotted!
1:12:20 sir, may I know the force equations of the system (3 masses and 4 springs) ? How to compute them?
Easiest way is to compute the kinetic and potential energy of the system and use the Euler-Lagrange equations: electron6.phys.utk.edu/PhysicsProblems/Mechanics/6-Oscillations/coupled-spring.html.
You can also use symmetry and conserved quantities to find the frequencies for many systems with little effort. The example of a linear triatomic molecule is a nice one. The center of mass undergoes uniform translation in one mode so a zero frequency mode exists; and the other two modes are easily visualized and the frequencies found.
Is derivation of relation in physics very important in technology?
I find application and use of law is the most important.Am I right?
Many of the most spectacular discoveries in Physics, at first, had NO connection with Technology. A striking example is Faraday's Law. *It's running our entire economy* When Faraday was interviewed the interviewer questioned the importance of his discovery and he asked: "what is the importance of your discovery?" Faraday answered: "The days will come that you will tax it."
Dear profecor EXCELLENT what I can not understand is how a wave with an open bountry does not dump to the end 9 the string and cames back , it is like hiting a rock to the midle of the ocean and the wave comes back to you strange .( 3 days wandered in my head )
www.acs.psu.edu/drussell/Demos/reflect/reflect.html. Open boundaries are a condition on derivatives while hard boundaries are conditions on displacements.
how can we understand the coupling between modes( more precisely anharmonic modes). how can we correlate it with spring mass system. i work with baw resonators.
it's all covered in my 8.03 lectures
Another amazing lecture professor! I have a question : why are we able to express any motion of the coupled osscilators as the superposition of the normal modes? I understand this somewhat intuitively but I don't feel I really know the reason.
it's the solution of the diff eqs. Also watch my 8.03 lecture on Fourier analysis.
thanks a lot for the reply professor I will
In fair chance, the metal ball will try to keep balance as you may see the metal ball those far away from the frequency source has more velocity. One thing right is the enery transfering from potentail to kinetic but since the frequency source is keep feeding more energy into the pendulum. Once the energy from the souce is stop then it will try to becomes equilibrium or you may see the multiples metal balls pendulum that some ball will not moving in some frequency and amplitude. But if you want all balls moving feed more energy let's swing over your head! In not a fair chance, where you are trying to send some energy to a nearest ball, in this case it will moving the nearest ball but if you provide more energy the other balls will keep balancing. That is why we are using long chain pandulum to at least absorb some effect of un-fair chance.
Question. I always get the steady state part of the equation correct. I always get the transient wrong. It's clear from my answers that i got part of it correct but, ultimately the answer for my transient solution is incorrect. I've watched your videos 8.03 lecture 5 and was hoping for an answer. You eventually said you'd not be giving the answer for the undetermined coefficients of the transient. I only read math books for fun so not an enrolled student. If you have any time could you give me a couple of most common mistakes? Maybe from that i can figure out what I'm doing wrong . Thank you!
I cannot add to the clarity of this lecture. I suggest you watch it again
Sir what is the proof that in coupling both pendulm can oscilllate with the normal mode
watch my lectures !!!!
Sir, quick question, how can we know when the system has an extra omega (omega++), since you did not mention the highest mode on your first example (two balls on pendulum connected with a spring)
how many minutes into the lecture?
2 balls on 2 connected strings have 2 resonance frequencies. n balls on n connected strings have n resonance frequencies. Watch my demo with n=3, very cool!
Lectures by Walter Lewin. They will make you ♥ Physics. At 1:04:18 you introduce the term highest mode but you did not mention that on 03:00 on your first example, my question is, why your first example doesnt have tht third mode (omega++)
Lectures by Walter Lewin. They will make you ♥ Physics. Oh thank you very much, I guessed that was it, I did, all of your demos are amazing, but by far the most thrilling one was the one with disproving the kirchoffs rule (8.02 lect 16) :)
2004? WHOOOAAAAAA
I was 3 years old.......then....
And finally coupled oscillators.........
Should I watch each video from before?
Or this as a standalone?
In all experiments the springs do not exert the repulsive force.
I can't figure out the mathematics but I absorbed the the visual-conceptual parts.
Screw it....I am going for the full course.....
I want to have better understanding of physics...and better Thinking ....
I will Start from your first problem sir....I couldn't do the Coupled oscillator one as well....
Sir at 11:47 why spring force is 2kx
THINKKKKKKKKKK
Sir I thought but didn't get
Help me......
I couldn't pretty much catch up upon the sow frequency and fast frequency.Can you please explain sir.
I cannot explain it any better than I did in my lecture.
Yes! I got it know.I re watched your lecture>>
I'm not able to figure out the force equation of a triatomic linear molecule in longitudinal oscillations. Please help.
use google
I tried googling, but all the sollutions use energy equations with complicated matrix solving methods. They do not use Newton's law to write the equation.
Lectures by Walter Lewin. They will make you ♥ Physics. wanted to know if it's possible to solve just using the kind of mathematical tools used in this video. With Newton's laws
@@rupeshknn cis.poly.edu/~mleung/CS4744/f03/ch04/linmol.html. The linear algebra makes your life easier; not harder. Lewin is doing the linear algebra without telling you that's what he's doing. Learn the Lagrangian approach. It's systematic and there is less room for error in getting the equations of motion and normal modes.
Sir what is reason that if there are 2 pendulm coupled together then then there will be two normal modes
Or
What is the reason that if there are n pendulm coupled together then there will be n normal mode
you will get n differential equations. To satisfy all you will need n normal modes.
@@lecturesbywalterlewin.they9259 sir why at normal modes the system resonates
solve your diff eqs !!!!
The number of vibration in per unit tine is called frequency
this lecture was recorded the on my birthday 😂😂
how is fx=2kx
how many minutes into the lecture?
12:32
2 pendulums - one displaced x to the right, the other displaced x to the left. Thus the spring force was 2kx.
ok thank you sir
sir why phase is zero because of less damping ?
how many minutes into the lecture?
An amazing lecture! The only thing I do not understand is why is there a pumpkin slice on his shirt
I think it is a slice of cantaloupe. My guess is he wears an homage to his breakfast to remind him what gives him the energy to perform his lectures.
Wat een prachtig college!
@Proffesor Lewin
I don't suppose you have access to demographic statistics regarding who is watching your lectures? I'm rather curious about who is making use of these, given I'm purely here for entertainment.
Yes I do have geographic statistics. Also age. Only 10% of my viewers are women.
Are you able and willing to release any of that data (age, location, gender)?
Ask your question and I will try to answer.
What is the average viewer age? Which 5 countries watch your lectures the most?
25-30
32:00
Hi Professor, and thanks for the wonderful lectures. Just a quick comment. At minute 22, in the second equation (x2), the trig equivalence should be minus 2sin half the sum/half the difference, right?
right!
@@lecturesbywalterlewin.they9259 Hello Prof. Lewin, may I ask then why in your lecture and French's book there is no negative sign in the equation of x2? Thank you in advance.
Thanks professor
You are welcome
Thank you sir.
Most welcome
To think that I was born the next day
Classes changed