I remember studying pendulums in my freshman physics course in college. What I don't recall is the solution being an intractable second order differential equation.
Very good example how to use units to find out equations is that I never remember ideal gas law. But looking at Bolzmann constant one can figure what multiplies and what divides.
This was explained really well. Seriously. I loved watching the notes style because it just makes my eye not get hurt. I hate looking just at documents in videos so, thank you.
I have found an interesting issue with classic pendulum equations: if you consider the Drag Force as the classic Stokes' force F=b*x' the pendulum eq. is (for some positive constants {a, b}): x'' +b*x'+a*sin(x)=0 a diff. eq. that under the transformation t -> -t is not time reversible! (which is commonly only atributed to entropy). But if instead the standard Drag Force F=b*(x')^2 is used, the diff. eq. becomes time reversible, but its solution are never decaying!... so it is needed to modify the Drag Force in something like F=b*x'*|x'| to recover the decaying solutions, so somehow the condition of been non-time reversible is required! But even with this improved drag force, solutions are never-ending in time, since the diff. eq. holds uniqueness of solutions due Picard-Lindelöf theorem... but, if I change the Drag Force by something like F=b*sign(x')*sqrt(|x'|)*(1+|x'|^(3/2)) which resembles Stokes' Law at non-near-zero low speeds, also the quadratic version at high speeds, but introduce a non-Lipschitz component at zero speed, the differential equation: x'' +b*sign(x')*sqrt(|x'|)*(1+|x'|^(3/2))+a*sin(x)=0 will be having decaying solutions that will achieve a finite extinction time t=T (so x(t)=0 exactly after t>T), also with a diff. eq. is not time reversible. Hope you can review this, is easy to see in Wolfram Alpha.
YEP ,GRAVITY IS AT THE CENTER OF THE EARTH , EVERY THING EVENTUALLY IS DIRECTED TO THE CENTER OF THE BLACK HOLE. ,ALL THAT TALK AND DIDNT GET ANY NUMBERS TO REALLY GET THE REAL ANSWER , YOU COULD OF AT LEAST OF HAD GIVEN US IGNORANTS SOME NUMERICAL EQUATIONS TO GO BY TO TRY TO UNDERSTAND .WE ARE IGNORANT BUT NOT STUPID ....NEXT TIME USE NUMERICAL DIRECTION , NUMBERS IS A LANGUAGE W ALL UNDERSTAND , LETTERS AND WORDS AND SIGNS IS A REPLACEMENT FOR WORDS ,ATYPE OF MATH (CALCULUS ) THAT SOME REALLY REALLY SMART PEOPLE DONT GRASP THA QUICKLY BECAUSE THERE NO NUMBERS TO YOUR THEORY ....GRAVITY ...
By the way, it just so happens that on Earth the time it takes for a pendulum to go one way is equal to the square root of its length. A 9 meter long pendulum takes 3 seconds to cross over. This is just T=2π(RootL/g) simplified.
In my opinion, this was the best concise explanation I found to help me understand a pendulum period problem involving differential equations. I am wondering if B = 0 (near the end of your explanation) does this merely confirm that the angle over time will tend toward the initial angle (theta sub-zero)?
Excellent videos. Thank you so much. I have started to watch each and every video of yours. I have one question about the speed of the oscillation, as the omega is in the denominator, a larger omega should only mean smaller speed right?I know you should be right, but I don’t seem to find the logic, can you please provide an explanation on why the speed is directly proportional to omega?.
Thanks Sathish! Omega determines the frequency of oscillations. A bigger omega means the pendulum is oscillating faster, meaning the period gets smaller
Hi Elliot. I have just started to see your videos and I want to say your work is amazing, thank for sharing it. Also, I would like to present a related topic for your videos: Recently, I have started studying the damped nonlinear pendulum dynamics, which is the simplest "realistic" physical model (since its consider friction energy losses), bit it is also known for not having yet any known closed-form solution. And when doing my research, I have found the paper "Finite Time Controlers" by V. T. Haimo, where continuous-time finite-duration differential equations are studied, and then I have realized three things: first, no linear differential equation could have finite-duration solutions, second, finite-duration systems' solutions are not unique, and third, there are aditional conditions to be fulfilled by the dynamics of the system to support finite-duration solutions, which as far I know, nobody in physics are taking in consideration when modeling physical phenoma (in the case of this video, is adding another point to be doing the problem: 1. Doing the Free Body Diagram, 2. Finding the sum of forces that equal zero (or equivalently, solving Euler equations for the Lagrangian), 3. Finding the solution to the equations, 4. Verify under which conditions the system support finite-duration solutions). Given this, since the system must be nonlinear to be realistic in the time variable (i.e., there exists an initial time and a final time for the experiment), I have start to wonder if the nonlinear damped pendulum haven't yet a known solution because nobody before have considered to pick finite-duration solutions (which also, cannot be analytic, since the only analytic and compact-supported function is the zero function, so standard power series expansions don't work - their domain must be choped at some points). Hope you review this issue in your videos to start to make wide known this issue.... I get stucked trying to find the solution by myself. Beforehand, thanks you very much.
Awesome video! I got a bit lost at 7:27; why do we need to look for a function whose second derivative is itself scaled by the extra factor omega squared?
Thanks Euan! That’s what we got by writing the F = ma equation for theta-it says that the acceleration of theta equals -g/l times theta (when theta is small, at least). So the second derivative of theta(t) is equal to theta again, but times that negative number -g/l, which is what I called -omega^2. That’s exactly the property of sin(omega t) and cos(omega t).
I am confused in a problem, what is the energy of a pendulum at its mean position when at rest and when oscillating? If the pendulum is hanging at some height above the ground. What will be the trajectory of the Bob if the string breaks at the time when the pendulum is passing through its mean position while oscillating? Thanks.
It's because the force points backwards, toward equilibrium. So when you pull the mass to the right, the force pulls it back to the left, and vice versa when you pull it to the left the force goes back to the right.
@@jovanmatic609 Backwards relative to increasing theta. Theta is zero when the pendulum is pointing straight down and increases in the positive direction when the pendulum is pulled to the right. Forces to the left will be in the direction of negative theta.
Is there a solution for bigger angles? I mean, without the hypothesis of small angles. If yes, is it a solution when the total energy is less than 2*m*g*L, and another one when the total energy is bigger than 2*m*g*L?
You can separate the energy conservation equation 1/2 m l^2 \dot{\theta}^2 - m g l \cos(\theta) = E in the form f(\theta) d\theta = dt, and then integrate both sides to get an equation like g(\theta) = t. This is a common route to finding trajectories---you do the integral to get g(\theta) = t, and then try to solve for \theta. In this case though the integral called an elliptic integral, and it doesn't have a simple expression in general (though a lot of properties are known about it), and moreover there's no simple way to solve for \theta(t).
PLEASE TRY TO COVER ALL THESE; Unit 1: Measurements and Experimental Analysis Units and dimensions, dimensional analysis Least count and significant figures Methods of measurement and error analysis for physical quantities Experiments based on using Vernier callipers and screw gauge (micrometre) Determination of g using the simple pendulum Young’s modulus by Searle’s method Specific heat of a liquid using calorimeter, the focal length of a concave mirror and a convex lens using the u-v method The speed of sound using resonance column Verification of Ohm’s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box Unit 2: Mechanics Kinematics in one and two dimensions (Cartesian coordinates only), projectiles Uniform circular motion, relative velocity Newton’s laws of motion Inertial and uniformly accelerated frames of reference Static and dynamic friction, kinetic and potential energy Work and power Conservation of linear momentum and mechanical energy Centre of mass and its motion; impulse Elastic and inelastic collisions Laws of gravitation Gravitational potential and field, acceleration due to gravity The motion of planets and satellites in circular orbits and escape velocity Rigid body, the moment of inertia, parallel and perpendicular axes theorems, the moment of inertia of uniform bodies with simple geometrical shapes Angular momentum, torque, conservation of angular momentum Dynamics of rigid bodies with a fixed axis of rotation Rolling without slipping of rings, cylinders, and spheres; equilibrium of rigid bodies The collision of point masses with rigid bodies Linear and angular simple harmonic motions Hooke’s law, Young’s modulus Pascal’s law; buoyancy Surface energy and surface tension, capillary rise, viscosity (Poiseuille’s equation excluded) Stoke’s law, terminal velocity, streamline flow, the equation of continuity, Bernoulli’s theorem and its applications Wave motion (plane waves only), longitudinal and transverse waves, superposition of waves Progressive and stationary waves The vibration of strings and air columns, resonance, beats The speed of sound in gases; Doppler effect (in sound) Thermal expansion of solids, liquids, and gases, calorimetry, latent heat Heat conduction in one dimension, elementary concepts of convection and radiation, Newton’s law of cooling; Ideal gas laws Specific heats (Cv and Cp for monatomic and diatomic gases), isothermal and adiabatic processes, the bulk modulus of gases Equivalence of heat and work, first law of thermodynamics and its applications (only for ideal gases) Blackbody radiation, absorptive and emissive powers, Kirchhoff’s law Wien’s displacement law, Stefan’s law Unit 3: Electricity and Magnetism Coulomb’s law; electric field and potential The electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field Electric field lines; flux of the electric field Gauss’s law and its application in simple cases, such as, to find field due to the infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell Capacitance, parallel plate capacitor with and without dielectrics Capacitors in series and parallel, energy stored in a capacitor Electric current; Ohm’s law; series and parallel arrangements of resistances and cells Kirchhoff’s laws and simple applications Heating effect of current
WOW, that video has somehow not sequestered the normal associated quality of image to sell us a premium "solution" to the quality problem they invented for us. Those type of things is criminal. Why does everyone let these things go on?! I mean, come on. I now feel I have to praise this one video fore not having a destroyed quality without the *"premium"* sh*t subscription. I hope the people that are now at the head of CZcams go to hell. Horrible, truly terrible.
I remember studying pendulums in my freshman physics course in college. What I don't recall is the solution being an intractable second order differential equation.
The equation is not intractable it actually has an exact solution in terms of Jacobi Elliptic Functions
Very good example how to use units to find out equations is that I never remember ideal gas law. But looking at Bolzmann constant one can figure what multiplies and what divides.
This was explained really well. Seriously. I loved watching the notes style because it just makes my eye not get hurt. I hate looking just at documents in videos so, thank you.
I have found an interesting issue with classic pendulum equations: if you consider the Drag Force as the classic Stokes' force F=b*x' the pendulum eq. is (for some positive constants {a, b}):
x'' +b*x'+a*sin(x)=0
a diff. eq. that under the transformation t -> -t is not time reversible! (which is commonly only atributed to entropy).
But if instead the standard Drag Force F=b*(x')^2 is used, the diff. eq. becomes time reversible, but its solution are never decaying!... so it is needed to modify the Drag Force in something like F=b*x'*|x'| to recover the decaying solutions, so somehow the condition of been non-time reversible is required!
But even with this improved drag force, solutions are never-ending in time, since the diff. eq. holds uniqueness of solutions due Picard-Lindelöf theorem... but, if I change the Drag Force by something like
F=b*sign(x')*sqrt(|x'|)*(1+|x'|^(3/2))
which resembles Stokes' Law at non-near-zero low speeds, also the quadratic version at high speeds, but introduce a non-Lipschitz component at zero speed, the differential equation:
x'' +b*sign(x')*sqrt(|x'|)*(1+|x'|^(3/2))+a*sin(x)=0
will be having decaying solutions that will achieve a finite extinction time t=T (so x(t)=0 exactly after t>T), also with a diff. eq. is not time reversible. Hope you can review this, is easy to see in Wolfram Alpha.
YEP ,GRAVITY IS AT THE CENTER OF THE EARTH , EVERY THING EVENTUALLY IS DIRECTED TO THE CENTER OF THE BLACK HOLE. ,ALL THAT TALK AND DIDNT GET ANY NUMBERS TO REALLY GET THE REAL ANSWER , YOU COULD OF AT LEAST OF HAD GIVEN US IGNORANTS SOME NUMERICAL EQUATIONS TO GO BY TO TRY TO UNDERSTAND .WE ARE IGNORANT BUT NOT STUPID ....NEXT TIME USE NUMERICAL DIRECTION , NUMBERS IS A LANGUAGE W ALL UNDERSTAND , LETTERS AND WORDS AND SIGNS IS A REPLACEMENT FOR WORDS ,ATYPE OF MATH (CALCULUS ) THAT SOME REALLY REALLY SMART PEOPLE DONT GRASP THA QUICKLY BECAUSE THERE NO NUMBERS TO YOUR THEORY ....GRAVITY ...
Great Explanation...specially the technique of figuring out the tangential component of mg. 🔥
By the way, it just so happens that on Earth the time it takes for a pendulum to go one way is equal to the square root of its length.
A 9 meter long pendulum takes 3 seconds to cross over. This is just T=2π(RootL/g) simplified.
so a 3 meter long pendulum takes one second? or is it a 1.5 meter pendulum that takes a second?
@@mateymate3066 A 3 meter long pendulum takes 1.7 seconds, the square root of 3.
@@mateymate3066 A pendulum of l = .25m has a period of 1 second (on earth). @woozy7405 “crossing” implies half the period.
In my opinion, this was the best concise explanation I found to help me understand a pendulum period problem involving differential equations. I am wondering if B = 0 (near the end of your explanation) does this merely confirm that the angle over time will tend toward the initial angle (theta sub-zero)?
thank you!
great! exactly how physics should be taught
Excellent videos. Thank you so much. I have started to watch each and every video of yours. I have one question about the speed of the oscillation, as the omega is in the denominator, a larger omega should only mean smaller speed right?I know you should be right, but I don’t seem to find the logic, can you please provide an explanation on why the speed is directly proportional to omega?.
Thanks Sathish! Omega determines the frequency of oscillations. A bigger omega means the pendulum is oscillating faster, meaning the period gets smaller
Great work!
Hi Elliot. I have just started to see your videos and I want to say your work is amazing, thank for sharing it. Also, I would like to present a related topic for your videos: Recently, I have started studying the damped nonlinear pendulum dynamics, which is the simplest "realistic" physical model (since its consider friction energy losses), bit it is also known for not having yet any known closed-form solution. And when doing my research, I have found the paper "Finite Time Controlers" by V. T. Haimo, where continuous-time finite-duration differential equations are studied, and then I have realized three things: first, no linear differential equation could have finite-duration solutions, second, finite-duration systems' solutions are not unique, and third, there are aditional conditions to be fulfilled by the dynamics of the system to support finite-duration solutions, which as far I know, nobody in physics are taking in consideration when modeling physical phenoma (in the case of this video, is adding another point to be doing the problem: 1. Doing the Free Body Diagram, 2. Finding the sum of forces that equal zero (or equivalently, solving Euler equations for the Lagrangian), 3. Finding the solution to the equations, 4. Verify under which conditions the system support finite-duration solutions). Given this, since the system must be nonlinear to be realistic in the time variable (i.e., there exists an initial time and a final time for the experiment), I have start to wonder if the nonlinear damped pendulum haven't yet a known solution because nobody before have considered to pick finite-duration solutions (which also, cannot be analytic, since the only analytic and compact-supported function is the zero function, so standard power series expansions don't work - their domain must be choped at some points). Hope you review this issue in your videos to start to make wide known this issue.... I get stucked trying to find the solution by myself. Beforehand, thanks you very much.
Thanks! That's a fairly technical problem, though I might talk about damped harmonic oscillators at some point
The formula for the period is valid for all angle? Or Just for little ones only
Thank you so much this explanation is great
Awesome video! I got a bit lost at 7:27; why do we need to look for a function whose second derivative is itself scaled by the extra factor omega squared?
Thanks Euan! That’s what we got by writing the F = ma equation for theta-it says that the acceleration of theta equals -g/l times theta (when theta is small, at least). So the second derivative of theta(t) is equal to theta again, but times that negative number -g/l, which is what I called -omega^2. That’s exactly the property of sin(omega t) and cos(omega t).
@@PhysicswithElliot Got it!! I missed that connection. Thank you for explaining and all the best with your channel 😁 Looking forward to more videos.
hey pendelum is what type of course
I am confused in a problem, what is the energy of a pendulum at its mean position when at rest and when oscillating? If the pendulum is hanging at some height above the ground. What will be the trajectory of the Bob if the string breaks at the time when the pendulum is passing through its mean position while oscillating? Thanks.
This video really helped me!
This is excellent.
from 4:50 this video projects away from highschool difficulty..
All concepts are highschool level though.
@@delete7316 Not the math part though
Nice!
I didn't expect to see you here! I'm loving the content from you and Elliot.
Excellently done. What is the software you use to write and draw? If it's ok for you to share with us.
Thanks Bobby! Procreate is the app
You can get an integral for t in terms of theta, although it isn't much easier to solve.
Why did you Take the sin of Omega and t instead of Theta?
How do you know to put MINUS sign in front of mgsin(theta)
It's because the force points backwards, toward equilibrium. So when you pull the mass to the right, the force pulls it back to the left, and vice versa when you pull it to the left the force goes back to the right.
@@PhysicswithElliot Backwards relative to what?
@@jovanmatic609 Backwards relative to increasing theta. Theta is zero when the pendulum is pointing straight down and increases in the positive direction when the pendulum is pulled to the right. Forces to the left will be in the direction of negative theta.
@@allyc1965 thanks so much!
Good
Is there a solution for bigger angles? I mean, without the hypothesis of small angles. If yes, is it a solution when the total energy is less than 2*m*g*L, and another one when the total energy is bigger than 2*m*g*L?
It can be using series expansion.
You can separate the energy conservation equation 1/2 m l^2 \dot{\theta}^2 - m g l \cos(\theta) = E in the form f(\theta) d\theta = dt, and then integrate both sides to get an equation like g(\theta) = t. This is a common route to finding trajectories---you do the integral to get g(\theta) = t, and then try to solve for \theta. In this case though the integral called an elliptic integral, and it doesn't have a simple expression in general (though a lot of properties are known about it), and moreover there's no simple way to solve for \theta(t).
Flammable Maths does a video on the complete solution. His presentation takes some getting used to, czcams.com/video/efvT2iUSjaA/video.html
PLEASE TRY TO COVER ALL THESE;
Unit 1: Measurements and Experimental Analysis
Units and dimensions, dimensional analysis
Least count and significant figures
Methods of measurement and error analysis for physical quantities
Experiments based on using Vernier callipers and screw gauge (micrometre)
Determination of g using the simple pendulum
Young’s modulus by Searle’s method
Specific heat of a liquid using calorimeter, the focal length of a concave mirror and a convex lens using the u-v method
The speed of sound using resonance column
Verification of Ohm’s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office box
Unit 2: Mechanics
Kinematics in one and two dimensions (Cartesian coordinates only), projectiles
Uniform circular motion, relative velocity
Newton’s laws of motion
Inertial and uniformly accelerated frames of reference
Static and dynamic friction, kinetic and potential energy
Work and power
Conservation of linear momentum and mechanical energy
Centre of mass and its motion; impulse
Elastic and inelastic collisions
Laws of gravitation
Gravitational potential and field, acceleration due to gravity
The motion of planets and satellites in circular orbits and escape velocity
Rigid body, the moment of inertia, parallel and perpendicular axes theorems, the moment of inertia of uniform bodies with simple geometrical shapes
Angular momentum, torque, conservation of angular momentum
Dynamics of rigid bodies with a fixed axis of rotation
Rolling without slipping of rings, cylinders, and spheres; equilibrium of rigid bodies
The collision of point masses with rigid bodies
Linear and angular simple harmonic motions
Hooke’s law, Young’s modulus
Pascal’s law; buoyancy
Surface energy and surface tension, capillary rise, viscosity (Poiseuille’s equation excluded)
Stoke’s law, terminal velocity, streamline flow, the equation of continuity, Bernoulli’s theorem and its applications
Wave motion (plane waves only), longitudinal and transverse waves, superposition of waves
Progressive and stationary waves
The vibration of strings and air columns, resonance, beats
The speed of sound in gases; Doppler effect (in sound)
Thermal expansion of solids, liquids, and gases, calorimetry, latent heat
Heat conduction in one dimension, elementary concepts of convection and radiation, Newton’s law of cooling; Ideal gas laws
Specific heats (Cv and Cp for monatomic and diatomic gases), isothermal and adiabatic processes, the bulk modulus of gases
Equivalence of heat and work, first law of thermodynamics and its applications (only for ideal gases)
Blackbody radiation, absorptive and emissive powers, Kirchhoff’s law
Wien’s displacement law, Stefan’s law
Unit 3: Electricity and Magnetism
Coulomb’s law; electric field and potential
The electrical potential energy of a system of point charges and of electrical dipoles in a uniform electrostatic field
Electric field lines; flux of the electric field
Gauss’s law and its application in simple cases, such as, to find field due to the infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell
Capacitance, parallel plate capacitor with and without dielectrics
Capacitors in series and parallel, energy stored in a capacitor
Electric current; Ohm’s law; series and parallel arrangements of resistances and cells
Kirchhoff’s laws and simple applications
Heating effect of current
Astounding video!
Thanks Raiyan!
Oops - something must be wrong!
Einstein said that time passes more slowly when there's more G! :)
good
Great vid
Also what is this software u used to write?
it feels so weird that I knew the answer for this: when a functions 2nd derivative is itself times -1 it's either sin or cosin
Holy W video
Hey professor! Great video. Did you really only begin your channel 4 months ago? I have good reason for asking, thx :)
I was lost at 3:10
ah, yes, one of the most known engineering theorems: sin T = T
why not start with ΣΓ=IΔ x α
WOW, that video has somehow not sequestered the normal associated quality of image to sell us a premium "solution" to the quality problem they invented for us. Those type of things is criminal. Why does everyone let these things go on?! I mean, come on. I now feel I have to praise this one video fore not having a destroyed quality without the *"premium"* sh*t subscription. I hope the people that are now at the head of CZcams go to hell. Horrible, truly terrible.
🤗🤗👍👌
I didn't understand anything. :(
Wait this isn't a video about Yugioh
Good video but you should not call it it “All about pendulums “ . This is not even fraction of the topic.