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# Physics Problem

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To make an argument that it does, we try breaking the shape down into many small squares of side length

To find the volume of revolution of the entire shape, we can find the volume of revolution of each small square, then add them all up.

The goal of this part of the problem is to write an equation expressing this "add up all the squares" process. We can do that with summation notation.

In this equation, we label the squares with a number:

When  for example, we look at this square:

That square has mass  $m_{17}$ and position vector $r_{17}$ The next square has mass  and position vector  etc.

The expression  means that we take  for each square and add them all together (using vector addition).

M is the total mass of a shape.

The part  means that there is some slight error when we approximate a shape by a bunch of squares, due to the edges having partial squares. If we cut the shape into smaller and smaller squares, the square shape gets closer and closer to the original shape, so the error gets smaller and smaller.

If we want to break the center of mass into components, we have

Build a summation for the volume of revolution of a shape revolved around an axis, assuming the shape is broken down into squares of side length S. The axis of rotation points in the y direction.

Sep 13, 2023

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The volume of revolution of a shape can be found using the disc method:

V = \int_a^b \pi [r(x)]^2 dx

where r(x) is the distance from the curve to the axis of rotation at x.

If we break the shape down into squares of side length s, we can approximate the volume of each square using the disc method:

V_i \approx \pi [r_i]^2 s^2

where ri​ is the distance from the center of the square to the axis of rotation.

The volume of revolution of the entire shape is then the sum of the volumes of all the squares:

V \approx \sum_{i = 1}^n V_i = \sum_{i = 1}^n \pi [r_i]^2 s^2

In the case of the circle, the axis of rotation is pointing in the y^​ direction. So, the distance from the center of each square to the axis of rotation is just the y-coordinate of the center of the square. We can write this as ri​=yi​.

Therefore, the summation for the volume of revolution of the circle is

V \approx \sum_{i = 1}^n \pi r_{xi} r_{yi} s

where n is the number of squares, s is the side length of each square, and (yi​,0) is the center of the $i$th square.

Sep 13, 2023