b. Find the center of mass if, instead of being constant, the ...

Question

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A thin plate of variable density delta of x is equal to 2 divided by the square root of x occupies the region bounded by the curve. Y is equal to 1 divided by x, and the x axis 41 is less than or equal to x and x is less than or equal to 4. Find the center of mass parentheses X Y bar of the plate. OK. So it appears for this particular prompt, we're asked to take all the information that is provided to us by the prom itself, and we're asked to ultimately find the center of mass of this thin plate. So now that we know that we're ultimately solving for the center of mass of this thin plate, we must first recall, based on our prior knowledge that the centroid parentheses X Y bar. For the lamina. Between Y is equal to f of x. And we have our X axis. With the variable density, and we need to recall and note that our variable density is delta. Of X, specifically our function delta of X. On our closed interval, because once again, as we should recall. Anything that's in square brackets represents a closed interval. So, on square brackets of A, B, our equation, or rather our expression, will be written as or will be given as X bar is equal to M subscript Y, which I'll call MY for short, divided by M. And Y bar will be equal to M subscript XMX divided by M. And we need to note where the moment about the y axis, which is denoted as MY, will be equal to the integral evaluated from A to B of x multiplied by FX multiplied by delta of XDX. And the moment about the x axis, which is denoted as mx, will be equal to 1/2 multiplied by the integral evaluated from A to B parentheses of F of X to the power of 2 multiplied by delta of xDX fantastic. So, moving right along here. So, we also recall our expression, or rather our equation for the mass, that's what lowercase m represents. Lowercase m denotes the mass, and the mass will be equal to the integral evaluated from B to A of. F of X multiplied by delta of XDX. And finally, let us state that Y is equal to FX and FX will be equal to 1 divided by X. So now, what we need to do is we need to solve for mass. So when we solve for M, the mass, we will find that the mass M is equal to the integral evaluated from 1 to 4 of 1 divided by X. Multiplied by 2 divided by the square root of x dx, which will be equal to 2 multiplied by the integral evaluated from 1 to 4 of 1 divided by. X to the power of 3 divided by 2. So it's gonna be x 3 divided by 2. Fantastic. And then this will be multiplied by DX. And this will be equal to 2, so once again it will be equal to 2 multiplied by the integral evaluated from 1 to 4 of x to the power of 3 divided by 2. DX, when we rewrite our expression. And this will be equal to 2 multiplied by x to the power of 1/2 divided by parentheses 1/2. So once again, it'll be equal to 2 multiplied by x 1/2 divided by parentheses of 1/22. And don't forget that we are evaluating this expression from 1 to 4. So this will be equal to -4 divided by the square root of x, evaluated from 1 to 4. Which will be equal to -4 divided by the square root of X, or rather, we need to plug in a value of 4 in for X, so it will be equal to -4 divided by the square root of 4 minus 4 divided by the square root of 1. And this will be equal to -4 divided by 2 + 4 divided by 1, which will be equal to -2 + 4, which will be simply equal to 2. So now, moving right along here, we now need to find the moment about the Y axis, so meaning we need to solve for MY now. MY is equal to the integral evaluated from 1 to 4 of x multiplied by 1 divided by x multiplied by 2 divided by the square root of x. The x, which is equal to 2 multiplied by the integral evaluated from 1 to 4 of x to the power of -1/22. DX. Which will be equal to 2 multiplied by x 1/22 divided by 1/22. Fantastic. And we need to note that we have to evaluate this expression from 1 to 4. So this will be equal to 4 multiplied by the square root of x evaluated from 1 to 4, which will be equal to 4 multiplied by parentheses of square root of 4 minus the square root of 1, which will be equal to 4 multiplied by parentheses of 2 minus 1. Which will be simply equal to 4 when we perform that expression. So, therefore, thus we could go ahead and write that X bar will be equal to MY divided by M, which will be equal to 4 divided by 2, which is simply equal to 2. So now we need to find the moment about the x axis, meaning we need to sulfur mx, and mx is equal to 1/2 multiplied by the integral evaluated from 1 to 4 of 1 divided by x in parentheses to the power of 2 multiplied by 2 divided by the square root of x dx. And this will be equal to the integral evaluated from 1 to 4 of 1 divided by x 2 multiplied by 1 divided by x 1/2. The x and this will be equal to the integral evaluated from 1 to 4 of 1 divided by x to the power of 5 divided by 2. The x which will be equal to the integral evaluated from 1 to 4 of x to the power of -5 divided by 2. The X, which will be equal to x to the power of -3 divided by 2 divided by -3 divided by 2, evaluated from 1 to 4. And this will be equal to -2 divided by 3 multiplied by x to the power of 3 halves, and we need to evaluate this expression from 1 to 4, which will be equal to drum roll, -2 divided by. 3 multiplied by 4 multiplied by the square root of 4 minus minus or negative. 2 divided by 3 multiplied by 1. So once again we're plugging in our variables for the evaluation. So 3 multiplied by 1 multiplied by the square root of 1. Fantastic. And this will be equal to when we perform this expression, or rather when we plug this expression into our calculator and we write it in its most simplified form, it will be equal to 7 divided by 12. So therefore, without a doubt, we could write that y bar is equal to mx divided by m, which will be equal to 7 divided by 12 divided by 2, which will be equal to 7 divided by 24. So that will mean therefore that the centroid. I, so once again, our synthroid will be written as parentheses of 2.7 divided by 24, and that is our final answer. Hooray, we did it. So going back up to the top to recap what we've learned, we could safely say that our final answer is once again, drumroll, 2.7 divided by 24. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.

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