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You don't need it. By using energy arguments instead of focusing on forces, you can get the work done by friction given only the initial height and the fact that the speed is constant! z

@ZaksLab I remember last year in the physics test there was a difficult question. This video made me remember the question. It was about friction. The text of the question is ( An object is sliding an incline surface as shown in figure, if the 20% of the surface is smooth and the other is rough with μ = 0.86 . if the object starts from rest and comes to rest at the bottom. Find The angle θ = ?)

You can start from 1 or 0 as long as your summation produces all the same terms. If we started this one from 0, we could use (-1)^n for the sign alternation (starts with + then alternates), then the rest of it would be x^(n+1)/(n+1). Check to see that the first few terms are the same, and in general when you're manipulating sigma notation, you're always kind of visualizing what its expanded form is going to look like.

you're welcome! z

you're welcome! z

Thanks! I do have the benefit of editing, but I've also been teaching for 20 years! z

I haven't solved that problem in a long time, and I'm not even confident the block would slide on a straight line trajectory during the motion . . . could it be a curved path? If you're just looking for final speed at the bottom, you should be able to get it with energy conservation and conservation of momentum (x direction). I have a note on my desk right now to make a video on that problem, but it will probably have to wait until the next time I teach mechanics (fall '25). z

@@ZaksLab Thanks I'll stay tuned - my physics 1 final was just today - would you believe fall '25 is exactly the semester I'll be taking 2nd year mechanics!

you're welcome! -z

finding minimum or maximum of a function: take the derivative and set it equal to zero. That's going to be the location of a max or min because it's the top of a hill or bottom of a valley where the slope is zero (calculus I optimization stuff), and you can usually infer which based on physical context. z

cool! z

thanks for your support! -Zak

you're welcome! z

I think you're right . . . chat gpt still makes a lot of mistakes with unusual physics problems. I write a ton of new problems for my classes every semester, and I try to push the boundaries into problems that are fairly original. If I check my solution against ChatGPT, it often makes several mistakes and needs to be "coached" through the solution, apologizing every time I point out an error. It's still a useful tool, but just try pointing out its mistakes to work toward a solution. z

that's an interesting idea! z

Glad I could help! z

glad I could help! note that if you are interested in creating math animations, you now have the secret to making polar curves spin: just plot r(theta-t), where t is the animation parameter, and you'll get a CCW rotation of the curve. z

thanks! z

thanks! z

This is a really good question, and one that bugged me for a long time. I know there are more rigorous ways to argue this, but let's simplify the problem and compare two simpler problems from calculus: finding the volume of a sphere using thin disks vs. finding the surface area of a sphere using thin ribbons. In the volume case, you can get away with using dz for the thickness of each disk, because the tiny amount you are cutting off the curved edge is negligible compared to the volume of the disk (volume of the disk is like pi*r^2dz, but the tiny angled ring we neglect is like 2pir*1/2*dx*dz, which has a product of *two* infinitesimals in it, making it of negligible order). In the surface area using ribbons case, we don't get so lucky: the surface area of a ribbon using dz for thickness is 2pi*r*dz, and the surface area using arc increment is 2pi*r*ds or 2pi*r*sqrt(dx^2+dz^2). This time the error committed is on the same order as the quantity being calculated (one infinitesimal), and that will lead to a finite cumulative error. With experience you just kind of develop a gut instinct for how detailed you need to be in order to not create a finite error -- generally speaking, surface integrals require the extra detail of using ds, and volume integrals can be treated more coursely, using dz for thickness. Try using just a dz for thickness in the surface area of a sphere calculation, and you'll see that it underestimates the total surface area compared to using ds. Hope this helps. z

@ZaksLab yes, it helps. Thank you!!!

you're welcome! z

you're welcome! z

thank you! z

This may not be a very satisfying answer, but friction is not doing net work on the system here. The reason why is that the contact point with the ramp is always stationary, so there is no mechanism for dissipating energy through slipping. Instead, the friction force is always a static friction force. But this force *is* exerting a torque on the system, and in fact it's critical to have this static friction force in order to enforce the rolling without slipping condition. The static friction force is the key to converting the rotational KE of the ball into additional height here: it supplies a torque slowing the rotation down, but simultaneously supplies a force pointing up the ramp. It is a weird thought to have friction acting as a conservative mechanism to convert one type of energy into another, but if you look at the force/torque solution of the problem, it does corroborate the answer given by energy conservation.

Thank you for your response! My another question is how would we solve the problem if the ball is allowed to both slip and roll?

@@huyviethungnguyen7788 that will depend on the angle of incline and the coefficient of kinetic friction. You'd have to do a force/torque approach to the problem instead of the energy approach.

@@ZaksLab Thanks a lot!

total distance! We found a_parallel, so we're describing motion along the hypotenuse.

you're welcome! z

Glad I could help! By the way, I recently updated this kind of problem with better production quality: th-cam.com/video/qTm3LyVVYI8/w-d-xo.html