Center of mass of a curved wire A wire of density δ(x, y, z)=15 √(y+2) lies along the curve 𝐫(t)

Center of mass of a curved wire A wire of density δ(x, y,...

Key Concepts

Center of Mass for a Continuous Object

The center of mass of a continuous object, such as a wire, is determined by taking a weighted average of the position vectors over the object’s domain. For a one-dimensional object, this involves integrating the product of the density function and the position vector along the curve, and then dividing by the total mass. This concept is fundamental in understanding balance and motion in systems where mass is distributed continuously.

Parameterization of Curves

Parameterization is the process of expressing a curve using a single variable, typically denoted as t, which maps to points in space via a vector function. This technique allows for the representation of geometric paths in a form that is convenient for performing calculus operations such as differentiation and integration. It is essential for setting up integrals required in physics and engineering problems.

Differential Arc Length

The differential arc length, often represented as ds, measures an infinitesimally small segment of a curve. It is obtained by differentiating the parameterized curve with respect to its parameter and taking the magnitude of the resulting velocity vector. This concept is crucial for converting line integrals from being with respect to the parameter t to being with respect to actual distances along the curve.

Line Integrals for Mass and Moment Computations

Line integrals extend the idea of integration to functions defined along curves. In the context of mass and moments, they are used to compute the total mass of a wire by integrating the density over the differential arc length, as well as the moments by integrating the product of the density and the position vector. These integrals provide a way to incorporate variations in density or geometry along the curve.

Density Functions in Continuous Mass Distributions

A density function describes how mass is distributed along an object and may vary from point to point. In calculations involving a wire or any continuous medium, the density function is integrated over the object's domain (using ds in the case of curves) to determine the total mass and weighted position (moments) necessary for finding the center of mass. This concept is key in handling non-uniform objects.

Transcript

00:01 Okay, folks, so in this video, we're going to take a look at problem number 34, where we're given a density function, and we're going to be looking for the center of mass of a wire lying along a curve, and then we're going to sketch the curve and the center of mass.

00:20 Okay, so first of all, let's just go ahead and sketch out the curve here.

00:27 The curve, this is the z -axis, this is the wire, axis and you'll see why i'm drawing it this way later.

00:36 So, you know, along the curve, we're given a function for y that parameterized by t and we're also given a function for z, which is also parameterized by t and y is t squared minus one and z is two t.

00:51 That means i can write y as z over two squared minus one because z over two is t.

00:57 So t squared is just z over 2 squared, and that is one -fourth, z -squarend minus 1.

01:03 So now you see why i'm drawing this y -and -z -axis like this, and that's because i want to write y as a function of z, because that's really easy to draw, because we have a parabola.

01:17 We have one -fourth and z -squarend minus 1, and the vertex of that parabola lies here.

01:23 This is negative 1, and you have a parabola that looks like this.

01:32 And because of the fact that our range is limited to negative 1 and 1 for t, so t is in the range of negative 1 and 1, that means that first of all, well, let's just plug it in here.

01:48 When t is negative 1, y is 0.

01:52 Okay, so that means our y cannot get greater than 0, and same as when t is equal to positive 1.

02:02 That's also when y is zero.

02:04 All right.

02:05 So we have graphed out the curve.

02:08 Now we can go ahead and look for the center of mass.

02:12 And as you recall, there's a formula for center of mass, and that formula is simply given by this.

02:23 It is a fraction, fraction, which is integral dm.

02:33 That's for the denominator, and the numerator is going to, be r dm okay that's that's really just the weighted sum of all the positions of the particles that constitutes the the object whose center of mass we're looking for anyway because of the fact that the curve r of t does not have an x component i can rewrite this center of mass vector in in this way so it's really just one over m, big m stands for the total mass, multiplied by sum of y had multiplied by, y, hat, multiplied by, let's see, y -d -m, plus z -hat multiplied by z -d -m.

03:29 Okay? so basically, we're going to have to do this integral, and then this integral, and then we're going to add them up, and divide.

03:37 By the total mass.

03:39 All right, anyway, the total mass, let's let's look, well, there's really a lot of integrals to do here.

03:44 We have three of them.

03:45 The total mass, let's do that first.

03:47 Total mass is really just density multiplied by ds.

03:51 Okay, and that, we know how to do it because we're given the density function, which is really just 15, y plus 2 multiplied by ds, between, well, first of all, this is equal to square root of y, plus 2 square root of 1 plus y prime of z squared multiplied by d z and between negative 2 and positive 2...

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