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.
Homework Statement
Let A be the region that in space bounded by the balls:
x^2 +y^2 + z^2 =1 , x^2 +y^2 +z^2 =4 , above the plane z=0 and inside the cone z^2 = x^2 +y^2 .
A. Write the integral \int \int \int_{A} f(x,y,z) dxdydz in the form:
\int \int_{E} (\int_{g^1(x,y)}^{g^2(x,y)}...
Homework Statement
A toy manufacturer wants to create a toy ice cream cone by fitting a sphere of radius 4 cm inside a cone with a height of 8 cm and radius of the base of 3 cm. The base of the cone is concave, but the rest of the cone is solid plastic so that with the sphere attached...
Homework Statement
A right circular cone has height 6 cm and base radius 2. It is over-filled with ice cream,
in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the
positive y–axis, and take the cross-section containing the x–axis. The top of this...
Homework Statement
A right circular cone has height 6 cm and base radius 2. It is over-filled with ice cream,
in the usual way. Place the cone so its vertex is at the origin, and its axis lies along the
positive y–axis, and take the cross-section containing the x–axis. The top of this...
Homework Statement
the task is to pump out the river from a cone with R radius and H height. How much will be the work (energy) used to do this? its a problem because I can't use here standard work definition W=\int F(s)\mbox{d}s, and I don't know what to do and how to start. please help:)
Homework Statement
I'm not from an English speaking country. I hope that you'll understend my question.
Cone made of steel is rotating on the wooden desk. Radius of cone is 4 cm, height is 10 cm.
Density of steel is 7,8*10^3 kg/m^3. Angular velocity of cone is 3.14 rad/s. Find the force of...
Homework Statement
Question B)
Homework Equations
The Attempt at a Solution
So I know the flux of S1 U S2 is the same as the flux for S1 + flux of S2, that is
Double Int of S(surface) = Double Int of S1 U S2 = Double Int S1 + Double Int S2
The problem I'm having is...
A plug in the bottom of the pressurized tank is conical in shape. The air pressure is 40kPa and the tank has a specific weight of 27kN/m^3. Determine the magnitude, direction, and line of action of the force exerted on the curved surface of the cone within the take due to the 40kpa pressure and...
Hi everybody,
Guys I'm a total stranger to physics. I need some help to find the relationship between the major/minor axes of an ellipse and the radius of a sphere in a cone of light.
For example, imagine a light source is located at 'h' height from a plane and a sphere(with a radius of...
Homework Statement
Evaluate the volume under z^2 = x^2 + y^2
and the disc x^2 + y^2 < 4.
Just wondering what I should write to constitute a proper solution. Would this do?:
V=(int)(int) z dA
R is {x²+y² < 4} [context: R in other problems was the region over which integrals were performed]...
Block inside a spinning cone -- Newton's 2nd Law problem
Homework Statement
See attempt at solution. I have attached everything there.Homework Equations
F_net = m * a_n (net force)
W = mg (weight)
a_n = 4*pi^2 * R * f^2 ... formula for "normal" component of acceleration where R = radius, f =...
Homework Statement
"A mass m = 16.0 kg is attached to the lower end of a massless string of length L = 71.0 cm. The upper
end of the string is held fixed. Suppose that the mass moves in a circle at constant speed, and that the
string makes an angle theta = 25deg with the vertical, as shown...
Homework Statement
I'm wondering whether or not it is possible to get the area of the base of an elliptical, non-right cone if the following two parameters are known:
- length from center of ellipse up to vertex
- angle that the sides make
Here is a simple visual...
Homework Statement
A particle is confined to move on the surface of a circular cone with its axis on the vertical z axis, vertex at the origin (pointing down), and half-angle a.
(a) Write the Lagrangian L in terms of the spherical polar coordinates r and ø.
(b) Find the two equations of...
Homework Statement
Hi, this isn't a homework question, it's for a physics project! Consider the following diagram:
I was wondering what the speed of the air would be at the point A (orange dotted line).
The red rectangle is a packet of air of height 1m and width d. Presuming that...
Homework Statement
"A conical surface (an empty ice-cream cone) carries a uniform surface charge \sigma. The height of the cone is h, as is the radius of the top. Find the potential at the centre of the top, taking infinity as reference point." - Griffiths
My result for the potential...
1. The problem statement
A truncated cone 30cm high is made of Aluminum. The dia at the top is 7.5cm, and 12.5cm at the bottom. The lower surface is maintained at 93 deg C, the upper surface at 540 deg C. the other surface is insulated. Assuming 1 dimensional heat flow, calculate the rate of...
Homework Statement
Find the volume, see attachment
Homework Equations
I can't find the proper equation for this cone.
The Attempt at a Solution
he triangle is a 3, 4, 5 triangle. Is the typical cone equation of 1/3*PI*r^2*h used or a different equation?
A forward light-cone is a surface that defines a region from which light cannot escape.
Similarly, a backward light-cone defines a region that light cannot enter.
What distinguishes these from event horizons?
[SOLVED] Electric Potential of Uniformly Charged Cone
I'm actually a senior in physics graduating this year, but wanted to review some E&M before grad school in the fall. I was apparently never assigned this problem during my sophomore year, but it's from Griffiths. I'm also aware that this has...
Hello
I am solving some problems from "Intro to EM" by David Griffiths ( third edition)
Problem 2.26 ( attached file 2.26.jpg) and I have also attached the solution from the solution
manual (griffiths-2.26.jpg). For both part a and b I am getting different answer.
I have chosen vertex as...
Homework Statement
Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 5.00 inches at the top and a height of 7.00 inches. At the instant when the water in the container is 4.00 inches deep, the surface level is...
Given the cone, z^2=x^2+y^2 z<= 1 filled with water, find the work needed to flip the cone upside down.
W = Fd
Well, I figured I could integrate the force along the distance (0,1) by multiplying the distance from 0 times the cross section circle area times the density. That would...
Homework Statement
Use a triple integral in rectangular coordinates to find the volume of the ice cream cone defined as follows
The region R in the xy-plane is the circle of radius 1 with center at the origin.
The sides of the cone are defined by the function z= \sqrt{x^2+y&2}
The top of...
Here are two pages showing examples of the Cherenkov radiation:
http://physics.syr.edu/hep/rich.html
http://www.iss.infn.it/webg3/cebaf/hadron.html
I don't understant why the cone is ahead of the particule path.
I thought that the cone of light formed behing the particule path (like the...
Hi all! I was trying to figure out how to find the volume of a cone with radius R and height h using integration with cylindrical coordinates. I first tried to set the the integral as:
\int_{0}^{2\pi}\int_{0}^{h}\int_{0}^{R}\rho d\rho dz d\phi
...but I think that this is setting up the...
I am looking for the formula to describe a spiral formed around a conical shape. If any particular details are needed, please make them variables and define them.
Thanks to all for the help!
I am trying to measure a concrete structure to compute the surface area.
I have included a sketch of the structure with the dimensions that I have measured. This is part of a drainage canal bank where two canals intersect. The banks along both canals are paved with concrete. I tried to...
Does the word cone more correctly refer to a surface (like the word sphere), or a solid (like the word ball); and if it refers to the surface, what would the solid be called?
Homework Statement
My AP physics teacher asked us to do an experiment on anything that involves a paper cone for my physics class, and I'm trying to think of something creative/original to do.
Homework Equations
No particular equations. As long as it's AP (12th grade) level physics.
The...
Homework Statement
In England, you can purchase fish and chips for a reasonable price. The reason it is so reasonable is because they give you no silverware, nor a plate. They just roll up a piece of paper in a cone and toss your food in. The vendors need to roll the cone in a perfect...
A reservoir in the shape of an inverted cone has a radius of 2 meters at the top and a depth of 6 meters. Wine is poured into the reservoir at a rate of 1 m^3/sec. At what rate is the depth of the wine increasing when the depth is 4 meters?
Help?
Homework Statement
A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.
The Attempt at a Solution
First I...
Homework Statement
Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. The height of the pile is increasing at a rate of ____ feet per minute when the pile is...
To calculate the centroid of a cone, it seems that you have to use calculus. It comes out to be h/4, where h is the height from the base of the cone. But intuitively I thought that the centroid would have been h/3 because that's a triangle's centroid, and the cone can be obtained by rotating a...
I was trying to understand Light cone but everytime i stuck somewhere or the other.Wiki has good description of it but can't provide enough information.Its a interesting subject ,i suggest going through it and obviously please help me out.
Homework Statement
With two fingers, you hold an cone motionless, upside down. The mass of the cone is m, and the static coefficient u. The angle of the tip, when viewed from the side, is 2θ. What is the minimum normal force required to hold the cone up (with each finger)? And, in terms of u...
Hi there.
I'm using the following equations as part of my physics research paper:
http://www.grc.nasa.gov/WWW/K-12/airplane/normal.html
The initial equation, concerning the half angle of the nose cone of the object, gives me a rather trivial answer at mach 1.5..
a > 0.199 (3sf)
Now...
Homework Statement
A particle of mass m moves on the inside surface of a smooth cone whose axis is vertical and whose half-angle is alpha . Find the period of small oscillations about a horizontal circular orbit a distance h above the vertex.Homework Equations
Not sure. Lagrangian maybe F = ma...
I'm machining a component as a means of testing one of our companies new machines. The objective is to manufacture a cone of maximum possible volume. The volume of a cone is given by: pir2h/3.
Given that h = 6 - r, how am i to calculate the maximum possible volume by means of integration...
What exactly is meant when people say that a Light Cone is "tilting"?
I understand the general idea of a light cone when it comes to how it's used to represent light particles. However, I do not understand what is meant when one states that in Relativity, "Light cones cannot be tilted so that...
Hello. I am having some trouble with the following problem and would be thankful if any of you could help me out.
Homework Statement
Let C be the hyperbola formed by intersecting the cone
x^2+y^2=z^2, z>0
with the plane x+y+z=1, and let
\textbf{f}(x,y,z)=<0,0,1/z^2>.
I am trying...
Homework Statement
Determine the centroid of volume for a right circular cone with base diameter of 100mm and an altitude of 200mm.
Homework Equations
I know that if the my xy-plane is parallel to the base of the cylindrical cone then the x and y coordinates of the centroid must be...
Homework Statement
Obtain an integral formula for the length of a curve p(theta) along a right cone. use spherical coordinates p and theta.
Answer: L = integral from -pi/2 to pi/2 of sqrt(p'^2 + p^2/R^2)d(theta)
Homework Equations
p is distance from origin
altitude a, radius 1...
Homework Statement
Find the centroid of the solid bounded below by the cone z = \sqrt{3(x^2+y^2)} and bounded above the sphere x^2+y^2+z^2=36.
Homework Equations
Let G be the given solid and denote its volume by V_{G}=\int \int \int_{G} 1 dV.
\frac{\bar{x}= \int \int \int_{G} x...
Homework Statement
Approximate this hill to a smooth cone with an elliptical base. find its volume by integration
Homework Equations
n/a?
The Attempt at a Solution
This hill is from a contour map and i have approximated the formula for the ellipse and the height.
i found the area...
My calculations show that until about 3.8 Gy ago (z ~ 1.7) the volume of space defined by our light cone had a high enough density to constitute a black hole, in the sense that 2GM > c2r. The further back in time, the more our light cone exceeded that threshold, because the density increases...
Hello all,
I'm trying to calculate the float line on a conical shaped vessel in water. The properties of the vessel are:
Weight - 10kg
Area of base of vessel - 0.311m2
I am using an equation I came across here:
http://answers.yahoo.com/question/index?qid=20080317200108AA5yNzY"...
you are given a contour map of a hill from which you are to approximate a cone and hence find volume.
each contour is an ellipse
my question: does centreing the ellipses effect the volume. I am pretty sure it doesn't but i want to verify
also. if i created 4 linear lines from the...