Cone Definition and 490 Threads

Homework Statement Homework Equations \frac{\partial\mathcal{L} }{\partial \phi} = \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{\phi}} The Attempt at a Solution z = r\cos\alpha s = r\sin\alpha v^2 = \dot{r}^2 + r^2 \dot{\phi}^2 sin^2\alpha \mathcal{L} =...

Homework Statement Water is leaking out of an inverted conical tank at a rate of 10,000 cm^3 / min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20cm/min...

Homework Statement Consider a truncated cone as shown in the figure. the material of the cone is a dielectric with top and bottom electrodes of different radii. Now a potential difference is applied across the capacitor - by connecting it to a battery - let's say. This creates an electric...

Homework Statement I need to find the volume of a cone covered with a plane z=h using multiple integrals. The scheme is something like this: Homework Equations Formula of the cone x^2-y^2-z^2=0 The Attempt at a Solution I tried to integrate ∫∫∫(x^2-y^2-z^2)dxdydz in all...

I would like to parametrize a skewed cone from a given vertex with an elliptical base, however I cannot seem to find the general formula for it. The parametrization given in http://mathworld.wolfram.com/EllipticCone.html produces a cone but not with the right vertex, ie, it is only a cone with...

Homework Statement This is a book problem, as follows: Find the volume between the cone x = \sqrt{y^{2}+x^{2}} and the sphere x^{2}+y^{2}+z^{2} = 4 Homework Equations spherical coordinates: p^{2}=x^{2}+y^{2}+z^{2} \phi = angle from Z axis (as I understand it) \theta = angle from x or...

If you want to know the value of the electromagnetic field at some point in space P at time t1, I assume that since EM is a relativistic theory, it should be possible to derive it using only the value of the field (along with charges, but let's say we are dealing with fields in free space) at an...

Have a fluid flow in a cylindrical pipe with insulation around it. The insulation is in the shape of a truncated cone. It starts at a thickness with a radius only slightly thicker than the pipe and the radius increases as you move along the pipe. The radius increases at a constant rate. The...

Homework Statement An inverted right circular cone of vertical angle 120 is collecting water from a tap at a steady rate of 18∏ cm^3/min. Find a) the depth of water after 12min, b) rate of increase of depth at this instant Homework Equations The Attempt at a Solution All...

I wanted to solve the problem of a cone rotating on its side over a table, around an axis that pass through it's apex, like in the figure. What I want to find is the angular speed ω, the spin of the solid, such that the cone "stands" over it's apex. I don't know how to set the condition...

To calculate the volume of the contents you use the formula for a cone, as long as the height of the product, h, is less than or equal to the height of the conical section, hcone. V=1/3rh2h if h ≤ hcone and rh is the radius at height h: rh=tan∅ if rh ≤ R. If the height of the stored...

Homework Statement Find a vector function that represents the curve of intersection of the two surfaces: The cone z = sqrt( x^2 + y^2) and the plane z = 1+y. Homework Equations z = sqrt( x^2 + y^2) and the plane z = 1+y. The Attempt at a Solution This problem can be solved as...

Is a cone a the degenerate of a 4 dimensional hyperbola? I only ask because I think it is and I am not sure. I am trying to get better at higher dimensional visualizations. My analogy being that a point is the degenerate of a 3 dimensional cone. With that logic wouldn't that make a cone...

Homework Statement This isn't a homework problem, but I got it off of an upperclassman's homework and decided to give it a shot. Consider a conical surface (like an empty ice-cream cone) with a height and top radius which are both h pointed up so that its axis lies along the z-axis and...

Find the volume of a cone with radius R and height H using spherical coordinates. so x^2 + y^2 = z^2 x = p cos theta sin phi y= p sin theta sin phi z= p cos phi I found theta to be between 0 and 2 pie and phi to be between 0 and pie / 4. i don't know how to find p though. how...

Homework Statement Find the geodesics on a cone of infinite height, x^{2}+y^{2} = \tan{\alpha}^{2}z^{2} using polar coordinates (x,y,z)=(r\cos{\psi},r\sin{\psi},z) with z=r\tan(\alpha) The Attempt at a Solution I am not sure with how should I expres the element dz^{2} ? When it is a...

I read that if the cone with apex angle 2α whose central axis is vertical, apex at the origin, then one can use spherical coordinate to calculate the solid angle of the cone ∫02∏∫0αsin\varphid\thetad\varphi However, what if the central axis is align to y-axis horizontally, instead of...

wrap a wire around a cone [urgent] [SIZE="3"][FONT="Arial"]Hello everyone, i need help solving a problem I'm facing so i can continue my project. So the problem I'm facing is that i got a cone 60cm height and 45 cm base diametre, and i want to wrap a wire with (0,3±0,1)cm diametre around the...

The problem statement, I know how to find the cg of a solid by using cross section but i just don't know how to find the cg of the cone by using the cg of a hollow cone for eg, we can calculate the cg of the half sphere by 1. calculating the cg of the hollow half sphere, than use it to...

It is stated that in the holographic principle (e.g., in http://en.wikipedia.org/wiki/Holographic_principle) that the the description of a volume of space is encoded on a light-like boundary to the region. But with respect to which position in the volume? In a black hole, it is clear, because...

Homework Statement A cone of height h and base radius R is free to rotate about a fixed vertical axis. It has a thin groove cut in the surface. The cone is set rotating freely with angular speed ω0 and a small block of mass m is released in the top of the frictionless groove and allowed to...

I am having trouble right now with the same problem (finding Ixx and Iyy). \begin{equation} I_{yy} = \int(x^2 + z^2)dm \end{equation} where \begin{equation} dm = \frac{2M}{R^2 + H^2} q dq \end{equation} and q is my generalized coordinate that is measured from the origin down the length of the...

Homework Statement Find the moment of inertia and center of mass of: A uniform cone of mass M, height h, and base radius R, spinning about its symmetry (x) axis. Homework Equations I = ∫R^2dm The Attempt at a Solution I tried using I =∫R^2dm, solving for dm I got dm=(M/V)dV...

Hello, I am supposed to find an expression for the general stress wrt height within a truncated cone of lower and upper radii a and b (a>b), pulled down from its vertical axis by a force of the same magnitude as that pulling it up. The diagram implies that the height between the upper and lower...

Homework Statement Homework Equations I'm guessing Stoke's Theorem? However, I'm not sure how to apply it exactly.. The Attempt at a Solution Looking at Stoke's Theorem, I'm still not sure how to apply it. I'm really just lost as to where to begin; is there even a \grad F to take? I know...

The problem gives a cone above a conducting grounded plane. (The xy plane) The cone has a voltage of 100V on it. It wants me to find the electric field between the cone and the plane. The angle the cone makes with the z axis is 10 degrees. And it is at a height of "h". So my method for...

if we cut a right cone parallel to the base having a two radius r and R The picture I want to use the volume of revolution around the y-axis we have the line y - 0 = \dfrac{h}{r-R} (x - R) x = \frac{r-R}{h} y +R The volume will be \pi \int_{0}^{h} \left(\frac{r-R}{h} y + R\right)^2 dy...

I am using the textbook called Classical Mechanics by John R. Taylor. Z = 1/M ∫ ρ z dV = ρ/M ∫ z dx dy dz On page 89, example 3.2, it says: "For any given z, the integral over x and y runs over a circle of radius r = Rz / h, giving a factor of πr2 = πR2z2 / h2." I wish the book would...

I am working independently from the book Precalculus Mathematics in a Nutshell by George F. Simmons. Although the book is fairly small, many of the problems are quite challenging, at least for me. I am stuck on this problem: "The height of a cone is h. A plane parallel to the base intersects...

Homework Statement Develop a formula for radius as a function of surface area for a cone with height three times its diameter. Homework Equations ∏rs + ∏r^2 = SA s = √(h^2+r^2) H = 3d or 6r The Attempt at a Solution Dont know what values to use for s = √(h^2+r^2).

How can one prove the formula for the surface of a cone as well as the volume of a cone without using calculus? Most of the online proofs use calculus. I ask this because these formulas are used in proving the formula for the volume of a solid of revolution and the surface area of a surface of...

Suppose there's a hemisphere of radius R (say) and a right cone of same radius R but ht. R/2 is scooped out of it then i have to find the center of mass of the remaining part. Here's how i approached... clearly by symmetry, Xcm = 0 Now, Let M be the mass of the hemisphere so...

Hello, I have been reading the excellent review by Eric Poisson, Ian Vega and Adam Pound:http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html In section 12, Eq.12.15, there's something that I don't quite understand. They write...

I have recently experimented with algorithms for rendering colour gradients. Linear gradients are no problem, but radial gradients have proved to be somewhat more difficult. A radial gradient focused at the centre is simply a matter of measuring the distance of a pixel from the centre and...

A book I'm reading (Companion to Concrete Math Vol. I by Melzak) mentions, "...any ellipse occurs as a plane section of any given cone. This is not the case with hyperbolas: for a fixed cone only those hyperbolas whose asymptotes make a sufficiently small angle occur as plane sections." It...

Homework Statement Find the volume of the region D in R^3 which is inside the sphere x^2 + y^2 + z^2 = 4 and also inside the cone z = sqrt (x^2 + y^2) Homework Equations The Attempt at a Solution So I decided that the best approach might be finding the area under the sphere and...

Hello, I have actually asked a similar question before, but I just realized something and I want to edit the question now: I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with infinitesimal height, and then adding up the lateral areas...

Hello, this is my first time posting on physics forums, so if I do something wrong, please bear with me :) I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with differential height, and then adding up the lateral areas of all of the...

Hi, I have a question about basic statics. I have heard from someone that the forces acting on a truncated cone in a hole of corresponding geometry is different from an ordinary block sliding down a wedge, since the normal force on one side of the cone will be affected by the normal force on...

Homework Statement Verify stokes theorem where F(xyz) = -yi+xj-2k and s is the cone z^2 = x^2 + y^2 , 0≤ Z ≤ 4 oriented downwards Homework Equations \oint_{c} F.dr = \int\int_{s} (curlF).dS The Attempt at a Solution Firstly the image of the widest part of the cone on the xy plane is the...

Homework Statement A right circular cone with radius r and height h is being filled with water at the rate of 5 cu in./sec. How fast is the level of the water rising when the cone is half full.Homework Equations V=r2h∏/3The Attempt at a Solution V=5t. The level of the water is determined by h...

I have to find the geodesics over a cone. I've used cylindrical coordinates. So, I've defined: x=r \cos\theta y=r \sin \theta z=Ar Then I've defined the arc lenght: ds^2=dr^2+r^2d\theta^2+A^2dr^2 So, the arclenght: ds=\int_{r_1}^{r_2}\sqrt { 1+A^2+r^2 \left ( \frac{d\theta}{dr}\right )^2...

Homework Statement Find limits of integration for volume of upside down cone with vertex on origin and base at z=1/sqrt(2). Angle at vertex is pi/2. Do this in cylindrical coordinates. Homework Equations None. The Attempt at a Solution My inner integral conflicts with the books...

Homework Statement This question comes out of "Introduction to Topology" by Mendelson, from the section on Identification Topologies. Let D be the closed unit disc in R^2, so that the boundary, S, is the unit circle. Let C=S\times [0,1], and A=S \times \{1\} \subset C. Prove that...

Hi everybody, I am trying to solve the following problem and I get stuck on the last question. I would appreciate a lot that someone helps me . Here is the problem: Let D be the region bounded from below by the cone z= the root of (x^2 + z^2), and from above by the paraboloid z = 2 – x^2 –...

I was taught that if you cut through a cone at any angle you will end up with a circle shape at the cones edge. Is this true?

> > > What is the maximum volume of a right circular cone with a slant height of \sqrt{3} \ units?Feel free to use Volume \ = \ \dfrac{1}{3}\pi r^2h.

Hello every one, This might sound as much stupid as it is confusing for me. Suppose the sun vanished right now (that would not happen practically, but I'm not concerned with that), then it will not be less than approximately a little more than eight minutes for us to know that the sun has gone...

How does circular motion in a cone without gravity work? If I have an object and I let it circle through the inside of a cone (with gravity) the two forces that act on the object are gravity and the support force. The resultant of the two forces lies on the plane of the circular motion and is...

Homework Statement A solid insulating cone has a uniform charge density of rho and a total charge of Q. The base of teh cone had a radius of R and a height of h. We wish to find the electric force on a point charge of q' at point A, located at the tip of the cone. (Hint: You may use the...