A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an object is more than "a mere geometric solid", but is "filled with, spread over by, or suffused with perceivable qualities such as color and warmth".The concept of surface has been abstracted and formalized in mathematics, specifically in geometry. Depending on the properties on which the emphasis is given, there are several non equivalent such formalizations, that are all called surface, sometimes with some qualifier, such as algebraic surface, smooth surface or fractal surface.
The concept of surface and its mathematical abstraction are both widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. The concept also raises certain philosophical questions—for example, how thick is the layer of atoms or molecules that can be considered part of the surface of an object (i.e., where does the "surface" end and the "interior" begin), and do objects really have a surface at all if, at the subatomic level, they never actually come in contact with other objects.
Homework Statement
It's open container with water which moves horizontally. It moves with some acceleration. Angle of incidence (α) and angle of reflection(β) on water are given (α=40 deg., β=20 deg.). We need to find container's acceleration.
Homework Equations
a=(v2-v1)/t (?)
The Attempt at...
I know that the surface roughness plays an role in the Magnus force exerted of a rotating cylinder. But, i cannot find an equation that includes the surface roughness in the equation of the Magnus force. If someone could state the formula (and preferably a source to read up more on it) it would...
Homework Statement
[/B]
Water rises in a capillary tube to certain height such that the upward surface due to surface tension is balanced by 7.5 X 10-4N
Force due to weight of the liquid. If the surface tension of water is 6 X 10-2Nm-1,the inner circumference of the capillary must be
Homework...
HELP I can't find the surface of revolution! By donuts I mean a circle that doesn't touch the axes (tore in french)
y^2+(x-4)^2=2^2 is my function ( y^2+x^2=r^2) and the axe of rotation is y
so y= sqrt(r^2-x^2)
the formula I know :
2* pi (Integral from a to b (F(x)*sqrt( 1+ (f``(x))^2))...
Homework Statement
We have an infinite cylinder that, from radius 0 to a, has a volume current density ##\vec{J(r)}=J_{0}(r/a) \hat{z}## , then from a to 2a, it has a material with uniform linear magnetic permeability ##\mu=(3/2)\mu_0##
, and at the surface, it has surface current...
Homework Statement
Find the centroid of solid enclosed by surface z= y^2 , plane x=0 , x = 1 and z =1 . The density is 1
Homework EquationsThe Attempt at a Solution
Here's my working .
Centoird = mass of inertia / mass
So , i find the mass first .
It's clear that the circle is on zx...
Homework Statement
Find the surface portion bounded by plane 2x +5y + z = 10 that lies in cylinder (x^2) +(y^2) = 9 ...
I have skteched out the diagram and my ans is 5sqrt(30) instead of 9sqrt (30) as given by the author ...
Anything wrong with my working ? Homework EquationsThe Attempt at a...
Homework Statement
Find the surface area of portion of plane x + y + z = 3 that lies above the disc (x^2) + (y^2) < 2 in the first octant ...
Homework EquationsThe Attempt at a Solution
Here's the solution provided by the author ...
I think it's wrong ... I think it should be the green...
Homework Statement
[/B]
Question :- Find the acceleration of block of mass ##M##. The coefficient of friction between blocks is ##\mu_1## and between block and ground is ##\mu_2##.
free body diagram at the end.
Variables :-
##f_1## - friction between blocks.
##f_2## - friction between block...
Homework Statement
Consider the metal-vacuum interface located at z = 0,the metal filling the entire half-space z ≥ 0, vacuum filling (!?) the half-space z < 0. The dielectric function in the metal in the long-wavelength limit is given by:
where ε0 is the vacuum permittivity. In the metal a...
Homework Statement
I'm just required to setup the integral for the question posted below
Homework EquationsThe Attempt at a Solution
So solving for phi @ the intersection of the sphere and the plane z=2:
z = pcos(phi)
2 = 3cos(phi)
phi = arccos(2/3)
so my limits for phi would go from 0 to...
Homework Statement
find the slope of tangent line to curves cut from surface z = (3x^2) +(4y^2) - 6 by planes thru the point (1,1,1) and parallel to xz planes and yz planes ...
Homework EquationsThe Attempt at a Solution
slope of tnagent that parallel to xz planes is dz/dy , while the slope of...
Hi. To frame things generally, I have a variable indoor light source that shines on an inclined plane. More specifically, I have a 3-D LED matrix that shines light onto a miniature pv array. Essentially, I am using an indoor “sun” to shine on the array. I measure the energy produced, and...
Hello everyone:
I studied in differential geometry recently and have seen a statement with its proof:
Suppose there is a Riemannian metric: ##dl^2=Edx^2+Fdxdy+Gdy^2,## with ##E, F, G## are real-valued analytic functions of the real variables ##x,y.## Then there exist new local coordinates...
Homework Statement
The electric dipole moment for the water molecule equals $$ p = 6.13 × 10−30 C · m $$ Suppose that in the glass of water all molecular dipoles could be made to point down. Calculate the resulting surface charge density at the upper water surfaceHomework Equations
[/B]
## P...
The surface of the Earth is moving approx. 1000 miles per hour at the equator. Which means, excluding natural wind and especially hurricanes, the air is also moving about 1000 miles per hour along with the surface, or the central areas of the Earth would be stripped of just about...
Hello bright minds,
I have a practical problem for which I need help solving, and I'm sure someone could help.
I am wanting to place a pick up (sort of a backwards facing venturi type structure) under my kayak to fill up and supply a box with fresh water for my live baits. I would like to know...
I've read that the surface area of an object in contact with the ground doesn't not affect the frictional force acting on it as it is pushed forward.
I kinda understand what is explained but I find it difficult to reconcile with what happens in real life...
Don't wheels reduce the surface...
Homework Statement
[/B]
The sphere of radius R has the potential at the surface equal to $$ V_0 = \alpha sin^2(\theta) + \beta $$ where ## \alpha, \beta ## are some constants. Find the potential inside, and outside the sphere.Homework Equations
$$V(r,\theta) = \sum_{l=0}^{\infty}(A_l r^l +...
Homework Statement
I have a block of mass m on a horizontal surface, which is covered in oil. The tell me the viscous resistance force is a function of the velocity, F(v)=-cv1/2, where I am assuming c is a constant of some kind. I need to find the time that it will take for the block to stop...
Homework Statement
What I basically want to do is to prove Gauss Law with a cylinder perpendicular to an infinite charged wire (I know I can do this simple, but I want to do it this way)
This is what I have done so far:
Homework Equations
$$\Phi=\int \frac{dq}{4\pi \varepsilon_0 r^2} \hat{r}...
Homework Statement
Here is a picture of the problem
https://www.dropbox.com/s/2bps6ga2o4hjpgw/hw4.png?dl=0
For those who don't want to click the link, basically the problem wants me to calculate the surface brightness of a flat surface at any angle if the source function S = B(T), which is a...
Homework Statement
A thin spherical shell with radius R1 = 2.00 cm is concentric with a larger thin spherical shell with radius 6.00 cm . Both shells are made of insulating material. The smaller shell has charge q1=+6.00nC distributed uniformly over its surface, and the larger shell has charge...
Homework Statement
Suppose that a surface has an equation in cylindrical coordinates of the form ##z=f(r)##. Explain why it must be a surface of revolution.
Homework EquationsThe Attempt at a Solution
I consider ##z=f(r)## in terms of spherical coordinates.
## p cosφ = f \sqrt{(p sinφcosθ)^2...
Homework Statement
Gate AB is a quarter-circle 10 ft wide and hinged at B. Find the force F just sufficient to keep the gate from opening. The gate is uniform and weighs 3000 lbf.
Figure and solution attached.
Homework EquationsThe Attempt at a Solution
I've been able to figure out how to...
I found this on the Internet . The formula is
Surface Area = R^2 \displaystyle \int _0 ^ {2 \pi} \int _{0}^{\pi} \sin \theta d \theta d \phi
I'm wondering why the limit of θ is from 0 to π only ? why not from 0 to 2π ? Because it's a perfect sphere...
I read that niobium metal has a Cooper-pair density of about 10^22 per cubic centimeter. However, when a current flows through a superconductor my understanding is that it all flows near the surface, beginning at the London penetration depth, which is a very small distance.
So, let's say that...
Homework Statement
when we calculate the electric field due to a plane sheet or the magnetic field due to a wire,are we calculating it at a single point or the whole field due to the total wire?
Homework EquationsThe Attempt at a Solution
Hi, I was hoping I could get some things cleared up. Recently my Solid State professor mentioned that we could simply, from the chemical formula, predict where the band crossings are going to be. For example, take LaFeAsO. Since La has a valency of +3, Fe of +3, As of -3, and O of -2, he...
Hi I'd like to ask how you would put a droplet at high viscosity (say 2.5 million times more viscous than water) on a solid surface? The droplet would have to be small (volume less than 10uL, r<1.34mm)..
Thanks!
Homework Statement
[/B]The coefficient of friction between the each mass and the floor are μM and μm respectively. which system accelerates faster under the same F in case:
1) μM = μm
2) μM < μm
3) μM > μm
Homework Equations
Friction force: ##f=mg\mu##
The Attempt at a Solution
1) Both...
Homework Statement
How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick?
Homework Equations
Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So...
Homework Statement
4 m masses, μ is the coefficient of friction. what is the tension and what should the maximum μ be to allow acceleration.
Homework Equations
Mass-acceleration: F=ma
The Attempt at a Solution
$$\left\{\begin{array}{l} mg-T=ma \\ T-3mg\mu=3ma...
Homework Statement
[/B]A mass m lies on an inclined surface with equal coefficients of friction μ=μs=μk. the pulley has also mass m and the weight is 2m. what is the velocity after it has descended distance h and during how much time.
Homework Equations
Moment of inertia of a massive disk...
Homework Statement
[/B]
We are heating a semi-infinite slab with a laser (radius of a stream is ##a##), which presents us with a steady surface heating (at ##z=0##), everywhere else on the surface the slab is isolated.
How does the temperature change with time?
Look at the limit cases: at ##t...
Homework Statement
[/B]
Consider an isotropic, homogenous dielectric sphere of radius R and constant relative permittivity ε, also permeated by a uniform free charge density ρ. Give an expression for the electrostatic potential V at the centre of the sphere by line integration of the electric...
Assuming that a black solar panel has an efficiency of 20%. This energy could be used instead of burning oil and decreasing CO2 emission. But the black color also accumulates a significant amount of the solar energy into heat locally on planet surface.
Comparing this with a white panel (non...
If we apply an electrical insulating material on a bare conductor which carries 25 Kv AC Voltage. What would be the surface voltage observed on the insulating material? Is there any test/method by which surface voltage can be measured on the Insulating Material.
The Insulating material has...
Hey everyone,
I've been stuck on this one piece of HW for days and was hoping someone could help me.
It reads:
The surface area, A, of a sphere with radius R is given by
A=4πR^2
Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
Is there a potential on the inner surface of a charged spherical shell?
I know that there is no electric field on the inner surface, as shown by Gauss's Law, but that isn't enough information to say that the potential (V) there is zero since E = dV/dr, so V could be a nonzero constant.
If...
Homework Statement
Let S be the portion of the paraboloid ##z = 4 - x^2 - y^2 ## that lies above the plane ##z = 0## and let ##\vec F = < z-y, x+z, -e^{ xyz }cos y >##. Use Stoke's Theorem to find the surface integral ##\iint_S (\nabla × \vec F) ⋅ \vec n \,dS##.
Homework Equations
##\iint_S...
Homework Statement
Hi everybody! I'm currently training at surface integrals of vector fields, and I'd like to check if my results are correct AND if there is any shortcut possible in the method I use. I'm preparing for an exam, and I found that it takes me way too much time to solve it. I...
Homework Statement
what will be Lagrange,s equation of motion for a particle confined to move on surface of sphere whose radius is expanding such that
Homework Equations
Euler-lagranges equation of motion
d/dt(∂L/∂{dq/dt})-∂L/∂q=0
The Attempt at a Solution
Z=(R+R0e^at)cosθ...
I'm given this problem to solve with the assumption as stated above. The answers need not be logical as long as the linkage mechanism can be simulated. I've attempted the question using Mercedes Mono Wiper mechanism but only manage to cover 41% given the width of the wiper. May I ask if there...
My teacher told me the other day that a semiconductor does not have a fermi surface. I didn't understand this remark. As I understand it the Fermi Surface is just the surface in k-space spanned by the highest occupied energy levels. Surely in a semiconductor you will also have some highest...
Homework Statement
Hi, would like to get a feedback on my answer to this question, did I do it right?
In part 2 of the question is there another way to calculate distance with friction?
the question is:
a disc is moving on icy surface has an initial speed of 12m/s 42m until it stops
1. what...
Homework Statement
It is evaluating a surface integral.
Homework Equations
∫s∫ f(x,y,z) dS = ∫R∫ f[x,y,g(x,y)]√(1+[gx(x,y)]2+[gy(x,y)]2) dA
The Attempt at a Solution
I set z=g(x) and found my partial derivatives to be gx=√x, and gy=0. I then inserted them back into the radical and came up...
Homework Statement
Two pools A and B have exactly the same depth, but A has 10 times the surface area, both at the top and at the bottom. Find the ratio of the total pressure (a) at the top surface of A to that at the top surface of B
(b)at the bottom of A to that at the bottom of B.
Homework...