Surface of revolution

Author: f | 2025-04-24

A surface of revolution is a surface obtained by revolving a curve around a fixed axis. Surfaces of revolution have no volume, as they are completely hollow. An axis of revolution is the axis around which a function is revolved to obtain a surface or a solid of revolution.

Surfaces Surfaces of revolution Sweep surfaces Parametric surfaces

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Surfaces Surfaces of revolution Sweep surfaces Parametric

D2 to the instrument. The operator at the instrument constantly observes the end D2 through the sight E as the other operator traces the kboundary of the area. Resistance to the pulling out of the wire may be had by checking its unreeling by they handle F, but such handle is primarily intended for winding up the wire on the drum D. When the boundary or outline of the surface being measured has been traced by the end D2, the area of the surface may be read from the vernier and the graduations on the disk A, which indication is brought about as follows: As the tape or wire D1 is unwound from drum D, it causes revolution of shaft W, which in turn, through gears H1 H2, (or other gearing used) causes revolution of cam C. The revolution of cam C, by reason of the reception Iof point N in groove C1, causes the carriage G to shift parallel to shaft V. Now, the operator at the instrument is moving the sight E as he follows with his eye the end D2 on the end of the wire D1 and as the sight E is moved, the spindle X is turned, which causes turning of the pinion C and thereby causes turning of the disk B. The revolution of the disk B or any movement thereof is communicated to the shaft T by the friction wheel M and by the friction wheel L to the disk A. It will

Surface of revolution as an analytical contact surface.

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Approximation of surfaces of revolution by developable surfaces

I.e., $h , then ${v_0} = {R_e}\sqrt {\dfrac{g}{{{R_e}}}} \approx \sqrt {g{R_e}} $${v_0} = \sqrt {9.8 \times 6.4 \times {{10}^6}} = 7.92 \times {10^3}m{s^{ - 1}} \approx 8km{s^{ - 1}}$The orbital speed of the satellite is independent of the mass of the satellite. The orbital speed of the satellite depends upon the mass and radius of the earth around which the revolution of the satellite is taking place.Time period of a satellite: It is the time taken by satellite to complete one revolution around the earth and it is given by$T = \dfrac{{2\pi r}}{{{v_0}}} = 2\pi \sqrt {\dfrac{{{r^3}}}{{G{M_e}}}} = 2\pi \sqrt {\dfrac{{{{\left( {{R_e} + h} \right)}^3}}}{{G{M_e}}}} = \dfrac{{2\pi }}{{{R_e}}}\sqrt {\dfrac{{{{\left( {{R_e} + h} \right)}^3}}}{g}} $For a satellite orbiting close to the earth’s surface i.e. $h $T = 2\pi \sqrt {\dfrac{{{R_e}}}{g}} = 84.6\min $.Kinetic energy of a satellite$K = \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}\dfrac{{G{M_e}m}}{r} = \dfrac{1}{2}\dfrac{{G{M_e}m}}{{\left( {{R_e} + h} \right)}}$Potential energy of a satellite$U = - \dfrac{{G{M_e}m}}{r} = - \dfrac{{G{M_e}m}}{{{R_e} + h}}$Total energy (mechanical) of a satellite$E = K + U = - \dfrac{{G{M_e}m}}{{2r}} = - \dfrac{{G{M_e}m}}{{2\left( {{R_e} + h} \right)}}$For satellite orbiting very close to the surface i.e., $h then $E = - \dfrac{{G{M_e}m}}{{2{R_e}}}$ .Binding energy of a satellite${E_B} = - E = \dfrac{{G{M_e}m}}{{2r}} = \dfrac{{G{M_e}m}}{{2\left( {{R_e} + h} \right)}}$ Escape speed of a satellite is defined as the minimum speed which is required for a body to project itself from the surface of earth such that it escapes the earth’s gravitational field. Escape speed ${v_e}$ is given by${v_e} = \sqrt {\dfrac{{2GM}}{R}} $

Surface Area, Surfaces of Revolution - mathreference.com

By two parallel planes. Roundness or Circularity — All the points on a surface are in a circle. The tolerance is specified by a zone bounded by two concentric circles. Cylindricity — All the points of a surface of revolution are equidistant from a common axis. A cylindricity tolerance specifies a tolerance zone bounded by two concentric cylinders within which the surface must lie. Profile — A tolerancing method of controlling irregular surfaces, lines, arcs, or normal planes. Profiles can be applied to individual line elements or the entire surface of a part. The profile tolerance specifies a uniform boundary along the true profile within which the elements of the surface must lie. Angularity — The condition of a surface or axis at a specified angle (other than 90°) from a datum plane or axis. The tolerance zone is defined by two parallel planes at the specified basic angle from a datum plane or axis. Perpendicularity — The condition of a surface or axis at a right angle to a datum plane or axis. Perpendicularity tolerance specifies one of the following: a zone defined by two planes perpendicular to a datum plane or axis, or a zone defined by two parallel planes perpendicular to the datum axis. Parallelism — The condition of a surface or axis equidistant at all points from a datum plane or axis. Parallelism tolerance specifies one of the following: a zone defined by two planes or lines parallel to a datum plane or axis, or a

1.3: Surface area (surfaces of revolution)

H} \right)}}$The orbital speed of the satellite when it is revolving around the earth at height is given by${v_0} = \sqrt {\dfrac{{G{M_e}}}{r}} = \sqrt {\dfrac{{G{M_e}}}{{{R_e} + h}}} = {R_e}\sqrt {\dfrac{g}{{{R_e} + h}}} $$\left( {asg = \dfrac{{G{M_e}}}{{R_e^2}}} \right)$When the satellite is orbiting close to the earth’s surface i.e., $h , then ${v_0} = {R_e}\sqrt {\dfrac{g}{{{R_e}}}} \approx \sqrt {g{R_e}} $${v_0} = \sqrt {9.8 \times 6.4 \times {{10}^6}} = 7.92 \times {10^3}m{s^{ - 1}} \approx 8km{s^{ - 1}}$Time period of a satellite: It is the time taken by satellite to complete one revolution around the earth and it is given by$T = \dfrac{{2\pi r}}{{{v_0}}} = 2\pi \sqrt {\dfrac{{{r^3}}}{{G{M_e}}}} = 2\pi \sqrt {\dfrac{{{{\left( {{R_e} + h} \right)}^3}}}{{G{M_e}}}} = \dfrac{{2\pi }}{{{R_e}}}\sqrt {\dfrac{{{{\left( {{R_e} + h} \right)}^3}}}{g}} $For a satellite orbiting close to the earth’s surface i.e. $h $T = 2\pi \sqrt {\dfrac{{{R_e}}}{g}} = 84.6\min$Kinetic energy of a satellite$K = \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}\dfrac{{G{M_e}m}}{r} = \dfrac{1}{2}\dfrac{{G{M_e}m}}{{\left( {{R_e} + h} \right)}}$Potential energy of a satellite$U = - \dfrac{{G{M_e}m}}{r} = - \dfrac{{G{M_e}m}}{{{R_e} + h}}$Total energy (mechanical) of a satellite$E = K + U = - \dfrac{{G{M_e}m}}{{2r}} = - \dfrac{{G{M_e}m}}{{2\left( {{R_e} + h} \right)}}$For satellite orbiting very close to the surface i.e., $h then $E = - \dfrac{{G{M_e}m}}{{2{R_e}}}$ .Binding energy of a satellite${E_B} = - E = \dfrac{{G{M_e}m}}{{2r}} = \dfrac{{G{M_e}m}}{{2\left( {{R_e} + h} \right)}}$ For a point close to the earth’s surface the escape speed and orbital speed are related as${v_e} = \sqrt 2 {v_0}$ Questions:1. At what height from the surface of earth the gravitational potential and the value. A surface of revolution is a surface obtained by revolving a curve around a fixed axis. Surfaces of revolution have no volume, as they are completely hollow. An axis of revolution is the axis around which a function is revolved to obtain a surface or a solid of revolution. A surface of revolution is a surface obtained by revolving a curve around a fixed axis. Surfaces of revolution have no volume, as they are completely hollow. An axis of revolution is the axis around which a function is revolved to obtain a surface or a solid of revolution.