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Zak's Lab
United States
Registrace 17. 08. 2020
Physics and Math with an emphasis on WHY.
Zak's Lab is primarily a physics and calculus instruction channel with hundreds of short lecture and example videos currently covering:
Physics I and II (trig-based). This crosses over to AP physics.
Physics I (calculus based).
Physics II (calculus based) is in progress.
Calculus I is coming Spring '24.
Calculus II
Browse dozens of playlists organized into the corresponding college course category. To see the full collection of playlists, please click the "playlists" button, as they do not all appear on the cover page.
I also like making math animations and algorithmic art, and those each have their own playlists as well.
Zak's Lab is primarily a physics and calculus instruction channel with hundreds of short lecture and example videos currently covering:
Physics I and II (trig-based). This crosses over to AP physics.
Physics I (calculus based).
Physics II (calculus based) is in progress.
Calculus I is coming Spring '24.
Calculus II
Browse dozens of playlists organized into the corresponding college course category. To see the full collection of playlists, please click the "playlists" button, as they do not all appear on the cover page.
I also like making math animations and algorithmic art, and those each have their own playlists as well.
Calculate the moment of inertia of a right triangle by using thin rods (part 1)
We calculate the moment of inertia of a right triangular slab rotating about an axis colinear with one leg by breaking into thin rods that are each rotating about one end. Using the previously established formula for the moment of inertia of a thin rod rotating about one end: I=1/3ML^2, we set up a moment of inertia integral and find the moment of inertia of a right triangle is 1/6Mb^2, where b is the length of the right triangle using an axis colinear with the height of the triangle.
zhlédnutí: 37
Video
Non-constant acceleration find the equations of motion when a=2-0.1t.
zhlédnutí 70Před 16 hodinami
When the acceleration of an object is changing, we have to go back to first principles to find the equations of motion. So, we start with the derivative relationships between position, velocity and acceleration: v(t)=x'(t), a(t)=v'(t). Given a non-constant acceleration a=2-0.1t, we guess the antiderivative to find the velocity as a function of time, and that's v=2t-0.05t^2 C, where C is an arbi...
Find the net force from lift, drag and weight vectors on a golf ball.
zhlédnutí 132Před 16 hodinami
In this vector addition by components problem, we are given lift, drag and weight vectors on a golf ball, and we find the net force using component-wise vector addition. Find the components of the force vectors: To start things out, we have to use a bit of geometry to determine the missing angles in the force diagram. Next, we resolve the force vectors into components. Find the net x and y comp...
Free fall problem given final velocity, find the time and distance using down as positive!
zhlédnutí 40Před 18 hodinami
In this free fall problem, we are given the final velocity for a rock in vertical free fall, and we calculate the time it takes for the rock to get to this velocity. In the second part of the problem, we find the total distance the rock has fallen in the time we calculated. Using down as positive, we set the origin of coordinates at the position from which the rock is dropped from the cliff, an...
Using a pendulum to measure gravity (including propagation of errors).
zhlédnutí 126Před 18 hodinami
Given the length and period of a pendulum (with uncertainty), we calculate the acceleration of gravity using a simple pendulum AND we compute the uncertainty in g using propagation of errors. Recall the formula for the period of a simple pendulum is T=2pi*sqrt(L/g). We solve the period formula for g and get g=4pi^2*L/T^2. This allows us to calculate the acceleration of gravity using the length ...
![Integrate like a physicist: definite integral of sin(pi*x)cos(pi*x) on [-1,1].](http://i.ytimg.com/vi/ZnIHY2Ra7iM/mqdefault.jpg)
Integrate like a physicist: definite integral of sin(pi*x)cos(pi*x) on [-1,1].
zhlédnutí 42Před 18 hodinami
We compute the definite integral of sin(pi*x)cos(pi*x) on [-1,1], showing how to integrate like a physicist (in other words focusing on how to exploit symmetries). In the first method, we notice that sin(pi*x) is odd and cos(pi*x) is even. But the product of an odd and even function gives us an odd function. We recall that the integral of an odd function on an interval symmetric about the origi...
Integrating like a physicist! Definite integral of 2cos(pi*x) using symmetry.
zhlédnutí 14Před 18 hodinami
We use a symmetry argument to compute the definite integral of 2cos(pi*x) on the interval [0,2]. This is a sinusoidal function with period 2, and when we integrate a sinusoidal function on a whole number of periods, we are guaranteed to get zero by symmetry. Keep an eye out for this kind of symmetry whenever you find yourself computing definite integrals of sines and cosines, and you'll be inte...
How to find the derivative of ln(sqrt(x^2+1)) using the chain rule and log properties.
zhlédnutí 30Před 21 hodinou
We calculate the derivative of ln(sqrt(x^2 1)) in two different ways: first we use log properties to bring a factor of 1/2 out in front of the derivative and we find the derivative using the chain rule and log properties. Alternatively, if we don't notice the advantage of using log properties in this derivative, we can view it as a function of a function of a function, and apply the chain rule ...
Derivative of sqrt(x+1/x^3) using the chain rule.
zhlédnutí 46Před 21 hodinou
We apply the chain rule to take the derivative of sqrt(x 1/x^3). First, we rewrite the expression using exponents as (x x^-3)^(1/2), then using the chain rule, we differentiate the outer function to get 1/2(x x^-3)^(-1/2) then we tack on the derivative of the interior function which is 1-3x^-4. Now that we've got the derivative of the square root of x 1/x^3, the main challenge is to simplify th...
Find the derivative of e^(2x)*sinx using the product rule and chain rule.
zhlédnutí 32Před 23 hodinami
We calculate the first derivative of e^(2x)*sinx using the product rule and chain rule. We begin by using prime notation to write an in-between step: this is really useful if you feel like you're doing too much at once! So we apply the product rule to write down (e^2x)'sinx e^2x*(sinx)'. Now we compute the derivatives, and the first one requires that we apply the chain rule because e^2x is a fu...
How to find the derivative of the inverse cosine function (derivative of arccos(x)).
zhlédnutí 38Před 23 hodinami
We derive the formula for the derivative of the inverse cosine function y=cos^(-1)x, which we can also call the derivative of arccos(x). To start the derivation, we rewrite as cos(y)=x, then we take the derivative of both sides. Recalling that y is a function of x, we have to use the chain rule on the left hand side, and we get -sin(x)dy/dx=1. Solving for dy/dx (the derivative of the inverse co...
Derivative of the inverse sine function (derivative of arcsin(x)).
zhlédnutí 33Před dnem
We derive the formula for the derivative of the inverse sine function; in other words the derivative of arcsin(x). To get a handle on inverse functions, it's usually useful to give them a name, so we start by saying y=arcsin(x). So, y is the angle whose sine is x, which allows us to turn around this equation and write sin(y)=x. Next, we differentiate both sides to obtain d/dx(sin(y))=d/dx(x). K...
Why the derivative of the inverse tangent is 1/(1+x^2).
zhlédnutí 104Před dnem
We compute the derivative of the inverse tangent function, which we could also call the derivative of arctan(x). We begin by giving the function a name, so y=arctan(x). This arctan(x) means "the angle whose tangent is x", so we can turn this equation around and write tan(y)=x. Now we differentiate both sides, keeping in mind that y is a function of x, so tan(y) is a function composition that re...
Magnetic field at the center of a rotating disk. Integrate using rings to find the magnetic field.
zhlédnutí 65Před dnem
Given a uniformly charged rotating disk with total charge Q, radius R and angular speed omega, we compute the magnetic field at the center of the disk. Using our previous result for the magnetic field at the center of a rotating uniformly charged ring, we compute the magnetic field at the center of a uniformly charged rotating disk by breaking the disk into rings. We compute the magnetic field ...
Magnetic field at the center of a rotating uniformly charged ring.
zhlédnutí 77Před dnem
Magnetic field at the center of a rotating uniformly charged ring.
Find the magnetic field of a coaxial cable using Ampere's Law.
zhlédnutí 1KPřed 21 dnem
Find the magnetic field of a coaxial cable using Ampere's Law.
Average torque exerted by friction for a hollow ball spinning to a stop.
zhlédnutí 236Před 21 dnem
Average torque exerted by friction for a hollow ball spinning to a stop.
Find the magnetic field of a toroidal coil using Ampere's Law.
zhlédnutí 421Před 28 dny
Find the magnetic field of a toroidal coil using Ampere's Law.
Use the ratio test to show divergence of the series n!/10^n (result of the ratio test is infinite).
zhlédnutí 70Před 28 dny
Use the ratio test to show divergence of the series n!/10^n (result of the ratio test is infinite).
Find the magnetic field of a solenoid using Ampere's Law -- not hand-waving!
zhlédnutí 130Před měsícem
Find the magnetic field of a solenoid using Ampere's Law not hand-waving!
Thin rod physical ballistic pendulum problem using angular momentum and rotational kinetic energy.
zhlédnutí 126Před měsícem
Thin rod physical ballistic pendulum problem using angular momentum and rotational kinetic energy.
Ampere's law with non-uniform current density: magnetic field of a thick cylindrical shell conductor
zhlédnutí 116Před měsícem
Ampere's law with non-uniform current density: magnetic field of a thick cylindrical shell conductor
Prove divergence of the series (4+3^n)/2^n by direct comparison to a geometric series.
zhlédnutí 67Před měsícem
Prove divergence of the series (4 3^n)/2^n by direct comparison to a geometric series.
Magnetic field of a thick cylindrical shell conductor using Ampere's Law.
zhlédnutí 135Před měsícem
Magnetic field of a thick cylindrical shell conductor using Ampere's Law.
Integral of 1/(x^2+5)^2 using a tangent trig substitution.
zhlédnutí 188Před měsícem
Integral of 1/(x^2 5)^2 using a tangent trig substitution.
Integral test for convergence of 2/(1+n^2). Integral test with the inverse tangent function.
zhlédnutí 128Před měsícem
Integral test for convergence of 2/(1 n^2). Integral test with the inverse tangent function.
Find the amplitude and maximum speed of a mass on a spring given the total energy of the oscillator.
zhlédnutí 168Před 2 měsíci
Find the amplitude and maximum speed of a mass on a spring given the total energy of the oscillator.
Magnetic field for a wire with non-uniform current density (Ampere's Law example).
zhlédnutí 152Před 2 měsíci
Magnetic field for a wire with non-uniform current density (Ampere's Law example).
Using Ampere's Law to find the magnetic field inside and outside a wire.
zhlédnutí 114Před 2 měsíci
Using Ampere's Law to find the magnetic field inside and outside a wire.
Using the geometric series sum formula to find the sum of (1+2^n)/3^n.
zhlédnutí 162Před 2 měsíci
Using the geometric series sum formula to find the sum of (1 2^n)/3^n.
Thank you so much
A little click moment for Heisenberg uncertainty.
Beautiful! It helps me to understand the Fourier transform problems. It’s a “click” moment!
Great tutorials as always! Thanks!
The force components are nicely explained, but I am having trouble understanding the "lift force" in this system. Where exactly does it come from?
for a golf ball, that's caused by the bernoulli effect due to the rotation of the ball
don't ask me to compute it though! I can guess that it gets bigger if the spin rate is higher, but that's about it!
gracias bro
de nada --z
This doesn't seem right ? I input into my nspire and I get (ln(x^2+1)/2)+(π|x|/2)-xtan^-1(x). My problem is the same but with limits from 1 to √3 and when I F(b)-F(a) this integral I get a different answer from when I input the limits into the nspire calculated integral.
Try differentiating your expression to see if you can get back to the original integrand. I can't speak for the nspire, because I don't have one, but I just differentiated my final answer and got back to arctan(1/x)! This is a pretty exotic integral, and I started getting pretty creative on exams and quizzes during the pandemic (partly boredom, partly trying to break the symbolic integration utilities that students were using to skip the work). Maybe it's exotic enough to break the nspire? z
@@ZaksLab I redid the problem , and the way you did it seems solid , but when I input the limits into this one it gives me ln(2)/2+π(√3/6+1/4). I also rechecked it on my nspire and it gave me almost the exact , but it computed ln(2)/2+π(√3/6-1/4). I inputted this into my hw on Webassign and it took it. I'm not sure where the negative came from. Since I did as you did by pulling the negative sign out of the -x/x^2+1 dx which would cause the entire equation to become positive because the there was already a negative sign on the left side of the from the formula uv- v du. So not sure ..but thank for the reply
I understand this better,it is a must watch video if you're looking for this question 😊😊😊
I'm pleased with the video
❤ very good
thanks! z
It’s killing it’s amplitude
I used to do this all the time in AP Calc but would fail most questions bc I would completely miss the answer due to user error or the teacher would just say "you didn't show your work".
It's unfortunate that so many calc teachers insist on showing trivial substitutions! When you end up applying integral calculus in engineering and physics, you're expected to just see the chain rule backwards as a matter of mathematical fluency. As a calculus and physics professor, I try to emphasize this point as early on as possible -- introducing the "chain rule backwards" for simple integrals before introducing explicit u-substitutions (which sometimes are actually necessary). z
Babe wake up zak's lab uploaded a new video
Does the magnetic field depend on the axis of rotation
Definitely! If we didn't have symmetry here, we wouldn't even have a steady current for the Biot-Savart Law. I wouldn't know where to start! z
What will happen if we apply vertical force T such as Mg/2 < T < Mg? Will the rod stay inclined or fall off the ground? My analysis shows it should fall, but my experiment showed it didn't. I'm stuck
Where's derivation in description
Why are these functions sin or cos anything? Don't understand how these got entangled with trig.
Here's a wikipedia article on the geometry of the hyperbolic functions and how they are analogous to the trig functions sine and cosine. I should definitely learn more and make a video on this topic! en.wikipedia.org/wiki/Hyperbolic_functions#:~:text=In%20mathematics%2C%20hyperbolic%20functions%20are,half%20of%20the%20unit%20hyperbola.
nicee❤
Nice 👍
🎉thank you for explaining for us the way we can apply to get the electric field
Hello, this video solution was good, thanks for that. Can you find out the moment of inertia of a cone which is not solid and it has Q charge unfiormly distributed over the surface. It is a right circular cone of semi vertical angle theta and height h. I would appreciate it, if you do could explain it in the most simplest way, because I am just a student with very little knowledge of integral calculus. Thank you!
Thank you very much for your video, this helped my friend and me understand this problem :). You saved us a lot of pain.
A great introduction to cycloid curves. Thank you!!!
you're welcome! z
I didn't realise its a smart move, I thought im just complicatkng myself, feelin satisfied
This is an interesting example! Looking forward to more like this! Thanks!
I have only a few more magnetic field videos to do this month, then I'm back to mechanics for a while (and second semester calculus). There's a cool genre of magnetic field calculations where you can look at the fields generated by charged rotating bodies, so that's the last few videos to finish that module in my E&M course. I swear I'll have my six college courses "completed" in the next 3-4 years . . . that's the goal anyway. z
@@ZaksLab I’m your big fan. I enjoy whatever you posted and look forward to more! You did a great job. Keep going!
Great work, thanks sir🙏🙏
thanks so much!
glad I could help -- z
nice work, finally a video which clarifies the unit conversions.
glad I could help! z
Wow Concept 🔥🔥🔥
very nice effort..sir
thanks! z
How to quote measurement in physics
This one really took me a while to fully understand, as knowing how doesn’t mean understanding why.
Thank you for making this informative, concise, and neat video❤❤❤
my pleasure . . . this one's a lead-in to the new lab I developed for my fall mechanics class last year (calculate the velocity of a ball bearing based on the maximum angle for the pendulum after impact)
All of you wrong = lbs per gallon + size pipe at bottom + nock offf few lbs for length ran.... ALMOST all city water from a tower is 60 psi ... most rural deep wells 1 hp sta-rite .. pump 60 -90 lbs as cut on - off. ...way better pressure in a shower! what i need is SHTF senario = 12 volt pump fills the tower of a 250 gallon tank at 20 feet high and at bottom i iuse 1 inch pipe and pressure gauge ! what is my pressure please ???? im going hook up 4x toilets with it.. so not need much .. lol
The testing part is inspiring too.
As always, the best of the best! One example helps me to understand many things.
instresting concept
Super Simple explanation Nice
Thank you, very good.
you're welcome! z
I looked at the video and then at the comments, surprised that not many have found this amazing explanation!! I got a little bit of practice for myself with this video, and some good concept clearance. Keep up the Good work!!
Thanks! Hey, the video is only two days old, so there is still hope! z
That was exactly what I felt. Perhaps popular stuff is for average people, but the best of the best is only for a few people who are above average.
@@ZaksLab I agree! Lets see where it goes, although i encourage you to keep going and explore out your own style! xoxo
@@juniorcyans2988 Oh my! I'd agree but there are infact popular videos that are good for all kinds of people! I mean, what differentiates popular stuff and non popular stuff are the analytics, not the quality itself right? XD For example, how dumb things can be more popular than educational stuff!
What if the graph starts at a negative value and then goes up?
how do i find the direction?
Ideally, can this be considered a perpetual motion machine?
After watching your videos very casually without any stress. I just realized that what makes math and physics difficult is: It’s not that the concepts are difficult to learn and understand. In fact it should really be the easiest to learn for people. It may be difficult getting into it, but it gets extremely easy and fun once you really understand it. What makes studying difficult is the stress and pressure of failing the course (and the consequences of if you fail) which messes with your psyche. And the TIME CONSTRAINTS of doing the exam. And the other pressures you face outside of the classroom, family, money etc. You gotta be PRECISE with your calculations as well. Also there’s something wrong with our culture (look at all the distractions from the media, And in the backwards village that i come from, families are more focused on religion and gossiping on the failures of others rather than on improving themselves.), The food people are putting in themselves is garbage too. (this is global, they just eat chapati’s from where i come from ). We’re deliberately messing our brains up. No wonder math is difficult.
Explained better than a drunk isp
Great
Can someone explain to me why the average is 401 hertz when one sine wave is 440 and the other 442 hertz? I don’t know how 401 got there.
That my friend is an epic typo! I don't recall anyone pointing it out until now. Rest assured my QC department is much more robust these days. z
@@ZaksLab ooooh Okay, damn.. Well you’re welcome for pointing it out then!😅
very good visualisation, can you please send the animation video if possible, we want to use it for school project, will give all credits
Attendance sang ta BSEd-Science 2A.
ohhhhh you made my day thank you thank you very muchhhhhhhh
you're welcome! z