Annulus Volume of Revolution
The problem involves finding the volume of an annulus created by rotating a segment of a circle around the x-axis. The solution demonstrates the mathematical steps and reasoning behind the calculation of the volume. Understanding the concept of volume of revolution in this context is essential for solving similar mathematical problems.
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? Annulus Volume of Revolution NOT to scale. 13 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis.
R (mm) h (mm) 13 5 20 16 Hint: 25.5 22.5 37 35 72.5 71.5 17.4 12.6 40.9 39.1 13.6 6.4 16.9 11.9 15 9 21.8 18.2 24.1 20.9
? Annulus Volume of Revolution NOT to scale. ? The general case: ? ? ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis.
? Annulus Volume of Revolution ?? ? Solution ? ? ? To solve for the general case refer to the diagram above:
Solution Considering the volume of the elemental disk ring of width ?? we have: ?? ?2 2 ? ? = ? Utilising symmetry and simplifying we can change the limits such that: ??2 2 ? ? = 2? 0 Noting that: ?2+ ?2= ?2 and substituting for ? gives: ??2 ?2 2 ? ? = 2? 0 Noting that: 2+ ?2= ?2 and substituting for ? this simplifies to give: ??2 ?2 ? ? = 2? 0 w 3 x 3 Integrating gives: = 2 2 V w x 0 Substituting the limits into the expression yields: ? = 2??31 1 =4 3??3 3
Solution Considering the volume of the elemental disk ring of width ?? we have: ?? ?2 2 ? ? = ? Utilising symmetry and simplifying we can change the limits such that: ??2 2 ? ? = 2? 0 Noting that: ?2+ ?2= ?2 and substituting for ? gives: ??2 ?2 2 ? ? = 2? 0 Noting that: 2+ ?2= ?2 and substituting for ? this simplifies to give: ??2 ?2 ? ? = 2? 0 w 3 x 3 Integrating gives: = 2 2 V w x 0 Substituting the limits into the expression yields: ? = 2??31 1 =4 3??3 3
? =4 3??3 This result is somewhat surprising in that it makes no mention of the radius of the circle. How can this be? The width of each segment is the same but the curvature of the upper edge gets smaller as the radius of the circle increases. So the cross-sectional area reduces, but this area is swept through a further distance. The two effects cancel each other out exactly. Note also that when = 0 the volume of revolution becomes a sphere of radius ?.
Extension Locate the centre of area of the segment The volume of a solid with a regular cross section can be found by multiplying the area of the cross section by the length of the path taken by its centre of area. Since we can calculate the area of cross section and the volume of the solid we can easily determine the distance to the centre of area from the axis of rotation.
RESOURCES Use the spreadsheet to plot answers as they come in (for fun!).
? Annulus Volume of Revolution NOT to scale. 13 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 20 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 25.5 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 37 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 72.5 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 17.4 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 40.9 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 13.6 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 16.9 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 15 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 21.8 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
? Annulus Volume of Revolution NOT to scale. 24.1 ?? ? 24 ?? The figure shows a segment of a circle. The chord is parallel to the x-axis. Calculate the volume of the annulus created by revolving the segment 360 about the?-axis. SIC_6
