Conical Definition and 140 Threads

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points. If the enclosed points are included in the base, the cone is a solid object; otherwise it is a two-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the lateral surface; if the lateral surface is unbounded, it is a conical surface.
In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe.
The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.
In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and right means that the axis passes through the centre of the base at right angles to its plane. If the cone is right circular the intersection of a plane with the lateral surface is a conic section. In general, however, the base may be any shape and the apex may lie anywhere (though it is usually assumed that the base is bounded and therefore has finite area, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly.A cone with a polygonal base is called a pyramid.
Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.
Cones can also be generalized to higher dimensions.

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  1. M

    Conical diffuser calculation using Bernoulli's Equation

    Homework Statement The following information applies to a conical diffuser: Length = 750 mm Inlet diameter = 100 mm Outlet diameter = 175 mm Water flow = 50 l/s Pressure at inlet = 180 kPa Friction loss = \frac{k(V_{1} - V_{2})^{2}}{2g} where k = 0.15 Calculate the pressure at exit...
  2. Saitama

    Cone rolling on a conical surface

    Homework Statement A round cone A of mass ##m## and half-angle ##\alpha## rolls uniformly and without slipping along a round conical surface B so that its apex O remains stationary. The centre of gravity of the cone A is at the same level as point O and at a distance ##\ell## from it. The...
  3. I

    Rate of water through a conical cone, in order to find constant k

    Homework Statement [10marks] A water tank has the shape of a vertex-down right circular cone. The depth of the tank is 9 meters, and the top of the tank has radius 6 meters. Water flows into the tank from a hose at a constant rate of 14 cubic metres per hour, and leaks out of a hole at...
  4. MarkFL

    MHB Kendra N's question at Yahoo Answers regarding work done to empty a conical tank

    Here is the question: Here is a link to the question: Applying integration to physics and engineering? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  5. P

    The factor of tension and frequency increase in a conical pendulum?

    This, is just a concept question: My teacher gave the Answer has something like this: However I understand most of it, except the part where it says: What is happening in this? Where does the 4*pi^2*L*sin(theta) come from, and how is v found? why is f = 1/t?
  6. M

    How do you set up a differential equation for draining a conical tank?

    I am working on a paper at the moment which has to do with draining tanks. I have already set up a differential equation which explains the drain from a tank where the cross-sectional areas of the container and of the outflow are constants. But now I have to set up one which explains the drain...
  7. D

    Particle moving on a conical surface

    Homework Statement A particle moves under the action of gravity on a conical surface z^2 = 4(x^2+ y^2), z ≥ 0, where z is the vertical axis. For initial position r = (1, 0, 2) and initial velocity ṙ = (0, 2, 0) find the extremal values of z along the trajectory. Take g = 10. Homework...
  8. G

    Conical pendulum question - I really don't know what to do - ?

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  9. L

    Conical Pendulum, find tension, radial force, speed, period, and angular speed

    Homework Statement A metal sphere is attached to the end of a string and then set in motion such that it rotates in a horizontal circle as shown in the sketch. The metal sphere has a mass of 0.5kg and the radius of the circle is .2m. 1)Find the tension in the string 2)Calculate the radial...
  10. J

    Classic Related Rates: Sand Conical

    I've been trying to figure out where my mistake lies in the first solution. Some help would be appreciated. I did notice I got the same solution twice, so I assume I just calculated dr/dt twice and I need to use a different equation for dh/dt? Is dh/dt=(3/4)*dr/dt? 1. Sand falls from a...
  11. S

    Conical pendulum circular motion question

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  12. I

    How Do You Calculate the Swing Angle in a Conical Pendulum?

    Conical pendulum question help!? Homework Statement By resolving forces horizontally and vertically and using Newtons second law, find an expression for the angle swung out. You must use calculus where needed. Use this analysis to answer questions 1-3 Q1: Will a child swing at a greater...
  13. J

    Is there an instantaneous angular acceleration for a conical pendulm?

    For a conical pendulum, there is an instantaneous centripetal acceleration. Does this mean there is an instantaneous angular acceleration of the pendulum towards the center?
  14. M

    Conical Pendulum Problem: Mass, Length, and Revolutions Per Minute

    Homework Statement A particle of mass 6 kg is attached to the centre B of a light inextensible string of length 10m. One end of the string is attached to a point A and the other end to a point C which is distance 8m directly below A. The particle is moving in a horizontal circle, with both...
  15. T

    Conical Pendulum proof of constant frequency

    Homework Statement A pendulum consists of a particle of mass m at the end of a light rigid rod of length l, the other end of the rod being freely attached to a stationary point 0. Let e(t) be a unit vector pointing along the rod, so that the position vector relative to 0 of the...
  16. G

    Moment of Inertia of a Conical shell

    Homework Statement Given that the conical shell has uniform density and thickness is made of one sheet, has mass M, height h, and base radius R, derive the moment of inertia about its axis of symmetry. Homework Equations I = MR^2 for a ring about its central axis. I = ∫dI - my approach The...
  17. J

    Work required to pump water out of a conical tank

    Homework Statement Find the work required to empty a 10m high conical tank with a radius at the top of 4m by pumping the water out the top of the tank. The water level is 2m below the top of the tank. Homework Equations \pi r^2 Similar triangles The Attempt at a Solution...
  18. S

    How can the weight of the cone influence the work done?

    Homework Statement A great conical mound of height h is built by the slaves of an oriental moarch, to commemorate a victory over the barbarians. If the slaves simply heap up uniform material found at ground level, and if the total weight of the finished mound is M, show that the work they...
  19. J

    Two Masses on Same String - Conical Pendulum

    3.(a) A particle P, of mass 0.03kg, is attached to one end of a light inextensible string OP, of length 1 m. The other end of the string is attached to a fixed point O. The particle moves in a horizontal circle, with centre vertically below O, at an angular speed of 2 revolutions per second. The...
  20. J

    Torque and angular momentum of a conical pendulum

    Homework Statement A ball (mass m = 250 g) on the end of an ideal string is moving in circular motion as a conical pendulum as in the figure. The length L of the string is 1.85 m and the angle with the vertical is 37°. What is the magnitude of the torque exerted on the ball about the...
  21. S

    Mass conservation in a conical tank

    Good Morning to all I saw this problem in one of the courses that I am taking this semester. It is very simple, it consists of an open conical tank being filled in the upper part with an stream (which is assumed to be cylindrical) of water (flow Qi through an area Ai). At the bottom of the...
  22. M

    Optimization question - optimal conical container

    Design the optimal conical container that has a cover and has walls of negligible thickness. The container is to hold 0.5 m^3. Design it so that the areas of its base and sides are minimized. information : 1) areas of the sides = (pi) x r x s 2) areas of the base = (pi) x (r^2) 3)volume of...
  23. S

    Fluids - Conical vs Cylindrical Water Clock

    I am researching water clocks through history. At some point, it was realized that for the container the water drips from, a conical container with the hole at its point was superior to a cylindrical container with the hole in its side. Could someone explain to me why conical containers are...
  24. R

    FEA issue with workbench plus question on equation with conical shells

    Hi everybody, I am designing conical shells in the aim to better understand the launch vehicle adapter and what changing thickness of conical shells will do to buckling load and natural frequency. I have an issue with certain equations I have characteristic equations with a critical axial...
  25. G

    Solved: Conical Pendulum: Calculating Tension & Period

    Homework Statement Fig. 6-53 shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.012 kg, the string has length L = 0.86 m and...
  26. K

    Angular velocity of a conical pendulum in rpm

    Homework Statement A conical pendulum is formed by attaching a 0.900 kg ball to a 1.00 m long string, then allowing the mass to move in a horizontal circle of radius 20.0 cm . What is the ball's angular velocity, in rpm? Homework Equations v=\sqrt{}L*g*sin(\vartheta)*tan(\vartheta)...
  27. D

    Flow rate and velocity form a Conical tank.

    I am working on a Hydro project. Just can't get this right. I have a water tank (cone shaped) with Radius 60 Inches and height 120 Inches. The Tip of the cone has a nozzle with a 3 inch opening. If the Air pressure is ignored. What will be the flow rate of water and pressure. ?
  28. Z

    Deriving Formula for Radius of Conical Section at Height h

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  29. S

    How Do You Calculate the Volume of Liquid in a Partially Filled Conical Glass?

    This is a problem that my lecturer gave us in class and it has been bugging me ever since. I have been unsuccesful in finding or calculating a proper solution so I am hoping PF will be able to help... This is the Q: Let H be the height of a conical glass which is filled to a height h. Find...
  30. A

    Work required to pump water out of conical tank

    Homework Statement Find the work done in pumping all the water out of a conical reservoir of radius 10ft at the top and altitude 8ft if at the beginning the reservoir is filled to a depth of 5ft and the water is pumped just to the top of the reservoir. Homework Equations None The Attempt at a...
  31. T

    How Does Conical Chamber Shape Affect Sound Amplification?

    Homework Statement This is a problem I've been pondering, and I simply need some recommendations for texts to read on the topic. Suppose you have an enclosed air chamber with an acoustic source on one end (http://beckerexhibits.wustl.edu/did/images1/hatpatent.jpg" ). How can one predict...
  32. T

    How Can You Predict Sound Amplification in a Conical Volume?

    Homework Statement This is a problem I've been pondering, and I simply need some recommendations for texts to read on the topic. Suppose you have an enclosed air chamber with an acoustic source on one end. How can one predict the amplification (most important), distortion, and...
  33. J

    Conical Solar Tunnel to Redirect Solar Radiation

    Am looking to build a solar tunnel that would not only redirect solar visible light radiation, but also heat, with minimum losses. Basically need to deflect angle of incident radiation by around 30 degrees. Am considering an inverted truncated cone design, with a polished inner surface...
  34. H

    Conical Pendulum Radius Calculation

    Homework Statement This setup of a conical pendulum is actually at the end of crane which is rotating, however I do not think that this should effect the situation: I need to calculate the radius so that I can then add this to the radius of the crane which will give my actual required...
  35. I

    Potential difference on a conical surface

    hi i am posting a problem 2.26 from griffiths EM book third edition.i am also attaching the solution from the book's solution manual. in the solution, griffiths has taken the ring as the differential element. but i want to know if we can take the small rectangular patch on the conical surface...
  36. I

    Potential difference on a conical surface

    hi i am posting a problem 2.26 from griffiths EM book third edition.i am also attaching the solution from the book's solution manual. in the solution, griffiths has taken the ring as the differential element. but i want to know if we can take the small rectangular patch on the conical surface...
  37. S

    Related rates question conical reservoir

    Homework Statement A conical reservoir is 50m deep and 200m across the top. Water is being pumped in at a rate of 7000m^3/min, and at the same time, water is being drained out at a rate of 9000 m^3/min. What is the change in diameter when the height of the water in the reservoir is 20m...
  38. L

    Conical Pendulum: Find Angle, Tension, Maximum Rate of Rotation

    1.The string of a conical pendulum is 1m long and the bob has mass 100g. It rotates at 0.5 revolutions per second. a) Find the angle that the string makes to the vertical. b) Find the tension in the string c) If the maximum tension which the string can bear is 2N, what is the maximum rate...
  39. T

    Conical Pendulum Concept Questions

    Homework Statement Q1. I am currently doing a physics assignment where i must answer some concept questions about a conical pendulum. So here they are: Does the centripetal acceleration and/or the net force alter if the launch angle changes? Q2. Would the centripetal acceleration and/or...
  40. I

    Conical Pendulum with free sliding ring

    Homework Statement A particle of mass m is tied to the middle of a light, inextensible string of length 2L. One end of the string is fixed to the top of a smooth vertical pole. The other end is attached to a ring of mass m, which is free to slide up and down the pole. The particle moves in a...
  41. ?

    How to Calculate Water Level and Boat Speed in Conical Reservoir Problems?

    Homework Statement A draining conical reservoir. Water is flowing at the rate of 50 m^3/min from a shalloe concrete conical reservoir (vertex down) of base radius 45m and height of 6m. a. How fast (centimeters per minute) is the water level falling when the water is 5m deep? b. How fast...
  42. S

    Solenoid power and conical area

    Hello , I have been wanting to find out the how heat generation of a tubular solenoid (pull-type)actuator is calculated? Secondly, why do most solenoid design use a plunger with a flat face/conical surface area . If I am looking for 1mm of stroke.. and a force output of 2500N, what apex...
  43. T

    NEED HELP A ball swung in a conical pendulum (circular motion?) question.

    Homework Statement A .30kg ball is swung in a conical pendulum whose length is 95cm. If the string makes an angle of 22 degrees with the vertical, what are (a) the balls speed and (b) the tension in the string Homework Equations The Attempt at a Solution I drew a freebody...
  44. A

    Empty Conical Tank: Use Torricelli's Principle to Find Time

    Homework Statement Use Torricelli's principle to find the time it takes to empty a conical tank of circular cross section standing on its apex whose angle is 45° and has an outlet of cross sectional area 1.0cm². The tank is initially full of water and at time t = 0 the outlet is opened and...
  45. N

    Conical Pendulum | Physics Motion Laws

    http://img25.imageshack.us/my.php?image=77494342.jpg I can do the first part, however am unsure of how to deduce the motion cannot take place unless the inequality is satisfied. Can someone please explain this part? Thanks
  46. N

    Work required to fill a conical tank

    A right circular conical tank of height 3 feet and radius 1 foot at the top is filled with water to a height of 2 feet. Find the work required to pump all the water up and over the top of the tank. similar triangles : x=y/3 water 62.5 lb/ft^3 ?? am i setting this up right
  47. S

    Hydostatic pressure at bottom of a cylindrical and conical tanks

    Please help me with the following simple and dumb questions: First, assumptions: - Ignore atmospheric pressure. - The liquid in the tanks is water with uniform density = 1g/cc. - all tanks are sealed at the bottom and are vertically positioned. - cross sections of the tanks are circular...
  48. P

    Help with conical pendulum problem

    Homework Statement okay here is the problem: A conical pendulum is formed by attaching a 500g ball to a 1.0m long string, then allowing the mass to move in a horizontal circle of radius 20cm. What is the tension in the string? Homework Equations My professor gave a hint that said use the...
  49. D

    Shortest path on a conical surface (Variational Calculus)

    I'm supposed to find the shortest path between the points (0,-1,0) and (0,1,0) on the conical surface z=1-\sqrt {{x}^{2}+{y}^{2}} So the constraint equation is: g \left( x,y,z \right) =1-\sqrt {{x}^{2}+{y}^{2}}-z=0 And the function to be minimized is...
  50. C

    What Is the Ball's Angular Velocity in a Conical Pendulum?

    Homework Statement A conical pendulum is formed by attaching a 0.900 kg ball to a 1.00 m-long string, then allowing the mass to move in a horizontal circle of radius 10.0 cm. The figure (Intro 1 figure) shows that the string traces out the surface of a cone, hence the name. (Figure Attached)...
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