A thin wire has the shape of the first-quadrant portion of...

Key Concepts

Parametrization of Curves

This concept involves representing a curve using a parameter, typically to simplify the integration process. For example, a circle can be parametrized using trigonometric functions, which is particularly useful when computing quantities such as arc length or when pulling a density function into the integral. The ability to express coordinates in terms of a single parameter is crucial in evaluating line integrals.

Line Integrals for Mass

When calculating the mass of a thin wire or curve, the mass is found by performing a line integral. In this context, the line integral of the density function along the curve, combined with the differential element of arc length, gives the total mass. This method integrates the variable density over the length of the curve, accounting for variations in mass density along different points of the wire.

Center of Mass Calculation for a Curve

Finding the center of mass involves calculating the weighted average of the positions along the curve. This is achieved by computing separate line integrals for the moments (the products of the density function, coordinate values, and differential arc length) about each axis and then dividing by the total mass. This process yields the coordinates of the center of mass, which reflect the distribution of mass along the wire.

Density Function in Integration

In problems involving variable mass distribution, the density function specifies how mass is distributed along the curve. Integrating this function over the curve—multiplied by the differential arc length—ensures that the calculation of physical quantities such as the total mass and moments for the center of mass appropriately accounts for changes in density along the wire.

Transcript

00:02 In this question, a thin wire traces out the first quarter, or sorry, first quadrant portion of the circle of radius a centered at the origin, and we are asked to find the mass and center of mass of this thin wire given its linear density function rho of xy here.

00:28 And so recall that the the total mass is the sum of all infinitesimal linear densities multiplied by linear lengths, or arc lengths.

00:48 And so we have to evaluate this line integral, and that involves finding limits of integration, a parameterization, r of t of the curve c, and finding the magnitude of the derivative of that parameterizing function.

01:22 Now, it's quite common, and it's quite common to parameterize a circle using a cos function for this x coordinate, and a y, or and a sine function for the y coordinate.

01:41 And so i will do that as well.

01:45 Note that the radius is a, so we should multiply both of these by a.

01:59 And because the derivatives of cos and sine are negative sine and cos respectively, our magnitude is going to be the square root of the sum of the squares of these things, elements, so we would have a squared sine squared, which i will abbreviate, plus a squared cos squared, also abbreviated.

02:25 And by the pythagorean identity, which tells us that sine squared plus cos squared equals one, we get the square root of a squared, which we know to be the physical radius, so the absolute value, its absolute value is just a itself.

02:59 Oh, and of course we need a range for this parameterization of the parameter t, and according to this standard parameterization, the parameter represents the angle formed by that particular point with the x -axis, the general point with a value t.

03:25 And since our angle ranges from zero to pi over two, that's what we need.

03:42 So now substituting the parameterization into our scalar field, and also multiplying that by r, the magnitude of r prime, and so we get this integrand.

04:01 Now at this point i'm assuming that most of my viewers will have already seen many videos of explanations of how to solve integrals like this, so i will just skip to the final answer.

04:15 Now let's find the center of mass.

04:27 So recall that the x -coordinate of this center of mass is given by this formula, and the y -coordinate similarly.

04:35 And the only difference between them is the x has an x multiplied by the row, and y has a y.

04:49 And these are just the weighted densities integrated over the arc lengths divided by the masses, so some kind of like an average...

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