What is Cone: Definition and 511 Discussions

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. B Hydrostatic pressure on top of a cone filled with water

Does the cone shape and extra water below the colum of water of 1 cm increase the pressure/vacuum effect at "B" or "A"? Is it simular to an inverted cone dependant on angle of sides of cones, when looking for pressure at the bottom of a cone or cylinder? In simple terms, how? Basic maths only...
2. B Model rocket physics: Why doesn't the motor go through the nose cone?

For those who have built those small model rockets, please help me understand a very very simple physics problem that i for some reason do not fully get. Why is it that the motor tube does not have a retention ring on it, to prevent the motor from flying out the nose cone? i know it cant be...
3. I Calculate new height of truncated cone

Hi, Suppose you have a truncated cone filled water with the lower radius being R, and upper r (R>r), and the height is H. R, r and H is known so the volume, V, can be calculated using V=1/3*pi*H*(R^2+R*r+r^2). Now suppose you remove some water so that you end up with a lower volume, V1. The...
4. What is the minimum time at a given angle on a cone with constant size?

I tried to come up with expression of time in terms of φ, α, and some constant value of cone size that does not depend on angle (I used R as cone radius). I though I could use that expression to see at what angle time is going to minimum, but I came up with expression from which I can't make a...
5. Volumetric flow of a overflowing cone

I solved the case where m=0.99999. Then the height at which it overflows can be obtained with the equation, when points on the liquid surface are chosen. Then the cross-sectional area is given by the circumference of the circle times the height that the parabola reaches, that cross-sectional...
6. Troubleshooting Coordinates System from Cone

I tried using coordinates system from cone, but not got what actually want to get. Any idea from you will greatly appreciated. Thanks
7. Can you explain the correct way to choose axes for circular motion analysis?

Haii, I don't understand why I need to choose my n-t components in the direction of a circular motion and can't just use them with the n-axis along the rope and the binormal perpendicular to the surface.
8. B A cone of photons from one (of many seen objects) hits the retina....

why is the result not like a movie screen where you are projecting 300 films at the same time, over each other ? (I would get it, if each object only sent one discrete beam, the next object another one, and so on, but it is a cone, of equally strong photons, being projected everywhere, into your...
9. Seeking Guidance to Find Surface & Volume Bound Charges of a Half Cone

This was a trivial question I had (which I posted here on the PF EM Forum: https://www.physicsforums.com/threads/bound-charges-polarisation-of-a-half-cone.1015308/). As I received no response on the above link I decided to post the same as a self formulated HW problem. Below I have attached an...
10. Path on Cone Fig.1: Angle ACP is 90°?

Fig.1 Fig 2 (the net of the cone) Point C is the turning point. ##\phi##= 90°. I wonder why the angle ACP is 90°. Is this a coincidence, or the "wire of minimum length" has anything to do with this? (Though, I thought the minimum length of the path can be acquired if ABP is a right angle)
11. I Rindler Wedge: Timelike or Spacelike? Intuition & Light Cone

Intuitively, the Rindler wedge is timelike in Minkowski coordinates and an object crossing the horizon enters a spacelike region. This seems at odds with my understanding of the light cone where the 2 regions are reversed. I think this may be related to the signature of the metric but I'm not...
12. How can I locate the coordinates of the centroid of a cone in Z?

This is the picture of the problem. My solution is: I'm not sure if the limit is 0 to 2 or 0 to 4...
13. Solving an Equation: Is it a Paraboloid or Cone?

Good day while solving some integral I met with the following equation z=sqrt(2-x^-y^2) that looks like a paraboloid?! I thought first that it might be a cone! any insights? thank you!
14. What Causes Loss of Particles in a Magnetic Mirror?

If I am not mistaken (hopefully) then the highest loss through the mirror ends in a magnetic mirror happens with particles that have a higher parallel component to the field lines than a higher perpendicular (gyro frequency). In a magnetic mirror if we map these properties onto the real physical...
15. Find the center of gravity of a combined cone and cylinder

Could I please ask for help with the following: Given: The centre of gravity of a uniform solid right circular cone of vertical height h and base radius a is at a distance 3h/4 from the vertex of the cone. Such a cone is joined to a uniform solid right circular cylinder of the same material...
16. "Gabriel's Horn" - A 3-D cone formed by rotating a curve

Problem statement : We have the graph of the function ##f(x)## shown to the right. The function ##f(x) = \frac{1}{x}## and the domain of ##x \in [1,\infty)##. We have to find the volume and surface area of the 3-D "cone" formed by rotating the function about the ##x## axis. ##\\[10pt]##Attempt ...
17. Solving Sphere in Cone Problem w/Semivertical Angle

Help please - okay so I have a question and struggling here. I need to know the radius of the sphere and how much water it displaces. One sphere inside an inverted cone One sphere for which the maximum possible amount of water is displaced. The problem is I’m only given the height of the...
18. I Probability of finding a particle outside its light cone

Say we just created a particle (high probability of one-particle state), is the probability of a very far away detector getting triggered at the time of creation (probability of finding a particle outside of its light cone) zero according to QFT? Since we can detect particles and make...
19. Finding ds² on a Cone: How to Use Geodesic Equations for Parallel Transport

I am having too much trouble to solve this exercise, see: Using (R,phi,z) ub is the path derivative U is the path V is the vector $$V^{a};_{b}u^{b} = (\partial_{b}V^{a} + \Gamma^{a}_{\mu b} V^{\mu})u^{b}$$ $$U = (0,\theta,Z)$$ I am not sure what line element to use, i mean, a circle around a...
20. Intersection of a few surfaces

Summary:: Describe what the intersection of the following surfaces - one on one - would look like? Cone, sphere and plane. My answers : (1) A cone intersects a sphere forming a circle. (2) A sphere intersects a plane forming a circle. (3) A plane intersects a cone forming (a pair of?)...
21. Calculate the center of mass of a deformed hollow cone trunk

I couldn't make progress in this problem, I would appreciate some suggestion on how could I attack this problem. Thanks in advance!
22. Optimization problem - right circular cylinder inscribed in cone

Please I do not want the answer, I just want understanding as to why my logic is faulty. Included as an attachment is how I picture the problem. My logic: Take the volume of the cone, subtract it by the volume of the cylinder. Take the derivative. from here I can find the point that the cone...
23. Related rates: water level in a cone

Firstly I'm having trouble understanding what water level means. I tried a quick google search and got the following: " water level(Noun) The level of a body of water, especially when measured above a datum line. " That doesn't help me. Is water level the distance from the base to where the...

26. Three conceptual questions on centripetal acceleration in a cone

Given such a diagram as shown above, we know that the normal force must be mg/sintheta. How is this normal force greater than the gravitational force conceptually? Is it due to the horizontal traveling (which must have been started by someone exerting a force?) compressing the sides of the cone...

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28. B How to calculate the amplification of a cone

Some where I read as sound travel away from the source its volume become 50% less every time cross sectional area doubles. Like throwing a rock into the lake wave rings get larger and larger diameter as the wave travel away from the rock. It seems to me same thing should happen in reverse...
29. Moment of inertia of a cone -- Why do we divide the equation dI = dm*r^2 by two?

Why we divide the equation dI = dm*r^2 by two when we calcule the moment of inertia of a cone relative to its symmetry axis?
30. I Surface of a cone by integration

If i want to calculate the volume of a cone i can integrate infinitesimal disks on the height h of the cone. I was told that if i want to calculate the surface of the cone, this approximation is not correct and i have to take the slanting into account, this means that instead of...
31. Volume of an oblique circular cone

This is not homework. I have given myself two parameters; ##\theta##, and ##\alpha##. (see figure, it is a side view): The idea is to find an expression for the radius of the circles as ##x## varies on that line (figure), then sum up infinitely many cylinders of infinitesimal thickness. The...
32. Flux of Electric Field through a cone

Since there is no charge inside the cone, the total flux through its surface is zero, hence Ø(lateral surface)+∅(base surface)=0. But ∅(base surface)=E.πR².cosΩ, because electric Field is homogenous. But by the figure, Ω is just arctg(h/R). So Ø(lateral surface)=-E.π.R².R/√(R²+h²). This is not...
33. How can we find the center of mass of a solid cone?

is this method even possible? anyways here is my attempt Step1) y= 2H/3 ( H is the height of the cone) step 2) we take the density (ρ)= 3M/π R2 H. The problem i am facing is to Find "dm"
34. MHB How Do You Calculate the Slant Height of a Cone?

Hello, Could anyone help me understand the steps on the below questions? A cone has a total surface area of 300π cm² and a radius of 10 cm. What is its slant height? A cone has a slant height of 20 cm and a curved surface area of 330 cm2. What is the circumference of its base? I'd really...
35. Heat Flow Through a Truncated Cone

This problem seems best treated in cylindrical coordinates. There is azimuthal symmetry, and there is no heat loss or generation within the cone, so our thermal conductivity equation reads: $$\vec{q} = -k(\frac{\partial T}{\partial \rho} \hat{\rho} + \frac{\partial T}{\partial z} \hat{z})$$ We...
36. Thermal Resistance of a Hollow Circular Cone

We can write our radius as a function of the height, z, of our cone: $$R(z) = \frac{R_2 - R_1}{h} z + R_1$$ Where h is the height of our cone, ##h = \frac{L}{40}##. Our cross sectional area, $$A = 2 \pi R t$$ can then be written as $$A = 2 \pi t [\frac{R_2 - R_1}{h} z + R_1]$$ This I am all...
37. I Light Cone in Rindler Coordinates: Visualization & Reasoning

Im trying to visualize what form the light cones take in Rindler coordinates. Below is my drawing + reasoning. Is it right?
38. Determination of the velocity of a particle in a hollow cone

Hi, please could I ask for help with the following question: A smooth hollow circular cone of semi-angle α, is fixed with its axis vertical and its vertex A downwards. A particle P, of mass m, moving with constant speed V, decribes a horizontal circle on the inner surface of the cone in a plane...
39. A Identifying a surface as a cone

Hi, I’m using some CAD software trying to automate some surface identification, and I’m looking to find a way to identify whether a surface is a cone. I will have access to vertices and normals at discrete points on the surface, but it will be expected that the number of these points will be...
40. Variation of electric field and potential along the axis of a cone

Options are at the top of page as a) b) c) d) Answer may more than one. Now since 'a' is distance from the smaller surface of cone so as we move along the axis area will increase,So current charge density will decrease and as we know J=sigma E,E will decrease,but V will remain constant since...
41. Calculating the moment of inertia of Cone

I'm Summing the Inertia of "donuts" with width dr and radius - r. I'm also "flattering" the cone into 2D and considering that each donut has different mass - because of the different height - h so: dm = 3 m h / (pi R2 H) dr I = ∫ dm r2 = 3 m h / (pi R2 H) r2 dr from triangle similarities H/R...
42. The potential between a cone and a plate

Homework Statement [/B] There is a conducting cone with angle α placed so that its vertex is normal to an electrically grounded plate, but electrically insulated from the plate and kept at a constant potential V. Find the potential V and the electric field in the region between the cone and the...
43. Potential of a solid, double cone shaped charge distribution

Does anybody know if there is an analytical expression for the electrostatic potential produced by a charge distribution confined to a double cone shaped region. Think of a beam of charged particles converging to a focus and then diverging again. The total charge in each thin, cross-sectional...
44. How do I find the intersection of three cones?

Homework Statement tl;dr: looking for a way to find the intersection of three cones. I'm currently working with a team to build a Compton camera and I've taken up the deadly task of image reconstruction. Background Theory: https://en.wikipedia.org/wiki/Compton_scattering For a single Compton...
45. Electric Potential on the axis of a truncated cone

1.Data: We have an truncated cone with a volumentric charge density ρ, and it's uniform. The image show the truncated cone and show some info of the radios. 2. Question. We need to calculate the potential on the vertical axis. note: adding an image of the problem but it's in spanish, hope...
46. Wrapping a Ribbon Around a Cone

Greetings everybody. This is my first post and I am looking for help with a little math/geometry/engineering problem. This has been a real brain buster for my colleague and I the past couple days so I am hoping somebody can help. I am not sure if this is the best section for it, but it...
47. I Differential structure on a half-cone

Hi, consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...
48. MHB 214 related rates gravel cone

Gravel is being dumped from a conveyor belt at a rate of $30\displaystyle\frac{ ft}{min}$ and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is $10 ft$ high...
49. B Problem about a cone and plane cutting it

It has a straight cone and a plane that cuts it as shown in the image. What angle should the plane form with the base of the cone ?, so that the plane cuts the cone in 2 parts of equal volume.
50. I Calculate the Efficiency of a Winston Cone

How can I calculate radiation transfer efficiency of a Winston cone, assuming a constant efficiency for every reflection at 99% and that the source is perfectly diffuse and covers completely the wider entrance of the cone? Also, are there more efficient non-imaging radiation concentrators with...