An Elementary Treatise On Analytic Mechanics With Numerous Examples
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139 (3) If a plane area revolve round any axis in its plane, the volume generated is equal to the area of the revolving figure multiplied by the length of the path described by its ventre of gravity.
Let A denote the plane area, and let it be of constant thickness and density, then (3) of Art. i83 becomes JJy ^^ ^y I I dxdy or %ny I idK = ^tt f fy die dy, (sabstituting dA for dx dy), .-. 2nyA = nfy^dx, (2) the integral being taken for every point in the perimeter of the area ; but the second me...mber is the volume of the solid generated by the revolution of the area (Cal., Art.
303) ; and the first member is the area of the revolving figure, A, multiplied by the length of the path described by its centre of gravity, 2ny.
Cor. — If the curve or area revolve through aiiy angle, 9, instead of 2n, (1) and (3) beeionie dys = d J yds, (3) and 0yA = \ej^y^dx, (4) and the theorems are stiU true.
ScH. — If the axis cuts the revolving curve or area, the theorems still apply with the convention that the surface or volume generated by the portions of the curve or area on opposite sides of the axis are afleeted with opposite signs.
What to read after An Elementary Treatise On Analytic Mechanics With Numerous Examples?
You can find similar books in the "Read Also" column, or choose other free books by Edward a Edward Albert Bowser to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
139 (3) If a plane area revolve round any axis in its plane, the volume generated is equal to the area of the revolving figure multiplied by the length of the path described by its ventre of gravity.
Let A denote the plane area, and let it be of constant thickness and density, then (3) of Art. i83 becomes JJy ^^ ^y I I dxdy or %ny I idK = ^tt f fy die dy, (sabstituting dA for dx dy), .-. 2nyA = nfy^dx, (2) the integral being taken for every point in the perimeter of the area ; but the second me...mber is the volume of the solid generated by the revolution of the area (Cal., Art.
303) ; and the first member is the area of the revolving figure, A, multiplied by the length of the path described by its centre of gravity, 2ny.
Cor. — If the curve or area revolve through aiiy angle, 9, instead of 2n, (1) and (3) beeionie dys = d J yds, (3) and 0yA = \ej^y^dx, (4) and the theorems are stiU true.
ScH. — If the axis cuts the revolving curve or area, the theorems still apply with the convention that the surface or volume generated by the portions of the curve or area on opposite sides of the axis are afleeted with opposite signs.
What to read after An Elementary Treatise On Analytic Mechanics With Numerous Examples?
You can find similar books in the "Read Also" column, or choose other free books by Edward a Edward Albert Bowser to read onlineMoreLess
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