Using disks to derive the moment of inertia of a cone about its symmetry axis. скачать в хорошем качестве

Using disks to derive the moment of inertia of a cone about its symmetry axis.

Given a solid cone of mass M and radius R, we calculate the moment of inertia using disks. We are given the formula for moment of inertia of a disk: I=1/2*mr^2 because we are using disks to derive the moment of inertia of the cone. 00:00 We begin by visualizing a thin disk slice of the cone, and we label the dimensions of the thin disk: we give it a height of y, a radius of x and a thickness of dy. 01:18 A note on density: next, we give a quick note on the definition of density as rho=M/V, which can be turned around to get M=pho*V. More importantly, we can write down the differential version of this formula and get dm=rho*dV. 01:53 Summary of strategy, then find the mass of a single disk, dm: we find the mass of our disk by using dm=rho*dV. This requires finding the volume of the thin disk, which is area times thickness, or dV=pi*x^2*dy, so the incremental mass of the thin disk is dm=rho*pi*x^2*dy. 02:47 Moment of inertia of a single disk: Now we can find the moment of inertia of our disk by plugging into the given formula for moment of inertia of a disk: I=1/2*mr^2. This gives us an incremental moment of inertia contribution of dI=1/2*rho*pi*x^4*dy. This expression is problematic because it has two variables in it! 03:32 Relating x and y to write dm in terms of one variable: We have to transform our expression for dI in terms of a single variable, and we do this by investigating how the x and y values for a disk are connected. They are related by the line y=-h/R*x+h, and we solve for x and replace x with its expression in terms of y for dm. 04:36 Final setup of the moment of inertia integral: Now that we have dI enitrely in terms of a single variable, y, we use integration to sum up the moment of inertia contributions as y goes from 0 to h. We evaluate across the limits of integration to obtain the moment of inertia of a cone about its symmetry axis, but there's a catch! 07:09 Eliminating rho from the answer: We want the final moment of inertia for the cone in terms of the mass and dimensions of the cone. So, we need to replace the density rho with mass divided by volume. Using the fact that volume is 1/3*pi*R^2*h for a cone, we find an expression for rho. We sub this into the moment of inertia result and simplify to calculate the moment of inertia of the cone: 3/10*MR^2.