Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lyin

step 1 Okay, folks, so now we're going to be talking about problem number 42. This is problem number 42

Find the center of mass and the moments of inertia about the...

Key Concepts

Differential Mass Element

The differential mass element is a key concept in continuous mass distribution problems. It is defined as the product of the density function and the differential arc length element (or volume element in three dimensions). This allows the conversion of integrals over mass to integrals over the geometric parameter or spatial coordinates, forming the basis for further calculations such as those of mass, center of mass, and moments of inertia.

Moments of Inertia

Moments of inertia measure the resistance of a mass distribution to rotational motion about an axis. For a continuous body, the moment of inertia about a coordinate axis is determined by integrating the product of density and the square of the distance from the mass element to the axis. This concept is fundamental in analyzing the rotational dynamics of objects and is used in design and engineering to predict stability and rotational behavior.

Parameterized Curves and Arc Length

This concept involves representing a curve in space using a parameter, and computing properties such as the length of the curve via integration. In this context, the position vector defined as a function of a parameter is used, and the arc length element is derived by taking the magnitude of the derivative of the position vector. This is critical in problems where integration is performed over a path, as it gives the appropriate scaling for integrals along the curve.

Center of Mass for Continuous Mass Distributions

The center of mass is a point that represents the average position of mass in a body or system. For a continuously distributed mass along a curve or wire, it is found by integrating the position vector weighted by the density and divided by the total mass. This concept extends the idea of the discrete weighted average to continuous systems using integrals, providing insights into the balance point of a system.

Transcript

00:01 Okay, folks, so now we're going to be talking about problem number 42.

00:06 This is problem number 42.

00:08 In this problem, we're going to be looking for the center of mass and the moment of inertia of a thin wire lying along a very complicated looking curve.

00:18 And we're also given a density function, which is parameterized by the variable t.

00:24 Okay, so first of all, you know, there's going to be a lot of computations, and because of the fact that we don't have a lot of time, i'm just going to skip over.

00:31 A lot of the trivial and time -consuming computations and leave those to you.

00:38 And all i'm doing here for this video is giving you a rough idea of the strategy or an overall picture of what you should have in your mind when you're doing this problem.

00:51 But anyway, for this problem, first of all, we're going to be looking for the center of mass.

00:56 And as you know, center of mass is a vector.

00:58 It's not just the number.

00:59 It's a vector of three numbers.

01:02 So we're going to be doing basically three integrals for the center of mass and moment of inertia as well.

01:09 Moment of inertia, you're going to have to find three numbers for the moment of inertia and that corresponds to moment inertia about x, about y, and about the z axis.

01:23 Okay, so that's where we're going with this.

01:26 Let's go ahead and start writing out some steps.

01:28 We have, first of all, center of mass is going to be written as 1 over the total mass multiplied by integral r dm now r is obviously the vector x x y and z okay so this integral is really three integrals okay and i'm just writing it in a very compact way and the total total total mass let's go ahead and try to find the total mass here okay so for the total mass it's obviously, it's actually very simple.

02:04 It's delta ds, because this is dm, and if you do the integral, you get the total mass.

02:14 Anyway, i'm going to write delta as 1 over t plus 1, multiplied by ds, but there's another way to write ds, which is to rewrite it as an expression in terms of t.

02:34 So how do you write ds in terms of t? well, as you remember, ds can be written as dx squared plus dy squared plus dz squared, and this can be written as x dot squared plus y dot squared plus z dot squared d t, where i used the notation dot on top of a letter to represent time derivative.

03:03 And we're given the curve x as a function of t is just t and y as a function of t is something something and z as a function of t is another function it's a parabola t squared over two but anyway if you do the time derivative if you differentiate with respect to time all three of these functions and you plug them back in here you're going to you're going to have one plus two t plus t squared d t all right.

03:33 Now, this is going to be written as 1 over t plus 1, square root of t plus 1 squared d t from, let's see, from 0 to 2.

03:47 And if you crank out this integral, you're going to get 2...

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