You start of the video with a fantastic explanation using simple logic and English. Bravo and thank you. It's amazing how this is possible and how many physics/math educators miss this totally.
Incredible, I could not find anywhere through a google search, why we use the integral of x squared, y squared, etc for moment of inertia. Thank you so much. I wish this was the introduction to moment of inertia is statics.
Yet another way to come up with I is to look at the circular analog of kinetic energy and replace appropriate variables and define I. KE = 1/2mV^2. Replace V with omega * r and integrate dm over a region. Just define the mr^2 as a point mass of inertia and replace it with I.
Thank you for the refresher class! I am currently dealing with this issue at work when comparing the lightness of wheel/tire combinations vs. durability against off road terrain features.
I'm glad this helps with a practical problem like yours. BTW, the previous video on this channel is how to measure mass moment of inertia of a wheel: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-MnrCfa4DOvc.html
The tangential force is equal to mass times the velocity squared divided by the radius. Who remembers that v = omega * r? omega being the angular velocity. Etc. All of this is derived using the fundamentals of differential and integral calculus. About the only time things can get weird is when the accelerations aren't constant.
The ultimate mathematical definition of what is BALANCED inertia/INERTIAL RESISTANCE is F=ma AND E=MC2 (ON/IN BALANCE): UNDERSTANDING WEIGHT AS BALANCED AND INTERACTIVE INERTIA/INERTIAL RESISTANCE: Define “mass”. You cannot. BALANCED inertia/INERTIAL RESISTANCE is fundamental. Weight ultimately pertains to and involves what is BALANCED inertia/INERTIAL RESISTANCE. E=mc² is F=ma. Energy has/involves GRAVITY, AND ENERGY has/involves BALANCED inertia/INERTIAL RESISTANCE. “Mass"/ENERGY IS GRAVITY. (Weight necessarily pertains to distance in/of SPACE.) Gravity AND weight are characterized by electromagnetic/gravitational interaction. NOTE: “Inertia" is not properly defined AND understood. The new/correct designation is inertia/INERTIAL RESISTANCE. By Frank DiMeglio
Explain the principal of Mass Moment of Inertia for a rigid body. Define the radius of gyration. How would you change the axis of moment of inertia? Prove your answer by necessary equations and graphs.
torgue also equals FXr which is Fr sin(theta). Of course that will be the perpendicular component of the force, but if it's already perpendicular then we know we have Fr Sin(90) = Fr. F = moment/r.
Inertia is the analog to linear mass and it derived in a similar manner to how kinetic energy is derived. I actually attended Purdue back in the 80's, and I became an expert in math in physics in a period of about a month. I was really a strange looking guy back then, but prior to that experience, I was, believe it or not, a playboy hippie. Isn't that funny? Anyway, KE is derived by starting with a distance and taking derivatives and then integration to take the time out of the equation and express everything in terms of m, v, and distance which everybody represented with an S, and then applying Fig Newton's law ( I used to eat those during class. It was fun.) Really, yeah both energy and inertia are consequences of Newton's second law. If we're talking about linear, then utilize F dot S and use kinematics to show that F dot S equals change in energy in terms of mass and velocity. In an angular system you will utilize angular distance, arc length if you will, and solve in terns of theta in terms of the radius and the mass. After all, the inertia of a point mass is MR^2. Each MR^2 is replaced by I. That's your inertia for a point mass. To find the total inertia for a volume of a function, use the density function to manipulate the equation so that the volumes cancel and the R^2 and Mass are left over. That will sum up all of the moments over the region bounded by the volume.
Use vector analysis. I think it's weird how so many people know how to calculate the sliding force that will cause an object to freely slide down an incline, but most don't know about the impending force and the least force.
