Unit Definition and 1000 Threads

What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base...

So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...

My textbook states that all the 14 bravais lattices have one of the 7 types of parallelepiped (7 crystal systems) that can be used as an unit cell. However in case of 3D HCP textbooks don't talk about this parallelepiped, rather they consider a hexagonal prism as the unit cell. Why is it so...

I have came across a problem where each point of a surface parallel to the x-y cartesian plane and having it's normal along the z axis is having velocity along the z direction ##v_z## and there exists a velocity gradient across the plane (e.g ##v_z(x,y)## , After time ##\delta t## it is written...

Dear Everyone, I am having trouble with an exercise problem. Here is the problem: Dummit and Foote Ed.2 pg 231: "Let $R$ be a ring with 1. Prove that if $u$ is a unit in R then so is $-u$." My Attempt: Suppose $u$ is a unit in $R$. Then, from Prop 1 (4) (if $R$ has an identity, then the...

Since the metric is euclidean in coordinates ##(ict,x)## it can be drawn in a plane, but if the metric is ##diag(1,-1)##, can both axis still be drawn in a plane ?

The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction. The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current...

Problem: Prove that for any $x \in R^n$ and any $0<p<\infty$ $\int_{S^{n-1}} \rvert \xi \cdot x \rvert^p d\sigma(\xi) = \rvert x \rvert^p \int_{S^{n-1}} \rvert \xi_1 \rvert^p d\sigma(\xi)$, where $\xi \cdot x = \xi_1 x_1 + ... + \xi_n x_n$ is the inner product in $R^n$. Some thinking... I...

Hi all, I have a question which I think should be a simple solve for most people but I cannot seem to get my head around it. It is a common case in geophysical surveying that a magnetic field vertical gradiometer might be used. This instrument uses two magnetometers separated by a known...

Can you give details?

I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...

Hello, I get that both polar unit vectors, ##\hat{r}## and ##\hat{\theta}##, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point ##P## on the plane is solely given in terms of one of the unit vector, the vector...

Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...

Homework Statement A 0.54 kg block of ice is sliding by you on a very slippery floor at 2.1 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0034 seconds. The block eventually slides at an angle of 21 degrees from its original...

If I have a beam of protons that go into a block of water, how can I know how many photons will be produced along the beam path?. I'm assuming all the photons have the same energy. I know that the energy deposition will follow a Bragg Curve, and I think that energy deposition is probably...

Homework Statement During a heavy rain, a section of mountainside measuring 2.5 km horizontally, 0.80 km up along the slope, and 2.0 m deep slips into a valley in a mud slide. Assume that the mud ends up uniformly distributed over a surface area of the valley measuring 0.40km x 0.40km and that...

Homework Statement Is N/m^2 a si unit I know Newton is and meters but togeather are there or is m^2 Homework Equations The Attempt at a Solution N/m^2 m^2

Homework Statement For one of my assignments I’ve been ask about the relationship between percent impedance and per unit impedance. Homework EquationsThe Attempt at a Solution I know that the x% is determined in transformers by short circuiting the secondary windings and slowly increasing the...

I know that spin is a type of intrinsic angular momentum. For electron spin is (1/2)ħ . But unit of (1/2)ħ is J.s, which is not the unit of angular momentum. Can you please explain this discrepancy?

Homework Statement You arrange your climate control system so that as it pumps heat out of the cabaret air and thereby chills and dries that air. Part of the heat your system removes from the cabaret air goes to the outdoor air and part goes back to the cabaret air. It turns out that delivering...

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 3.1: Manifolds ... I need some help in order to understand Example 3.1.3 ... ... Example 3.1.3 reads as follows:In...

Homework Statement Let ##S=\{a+bi+cj+dk \mid a,b,c,d \in \mathbb{Z}\}## be the ring of integral Hamiltonian quaternions, where multiplication is defined using the same rules as in ##\mathbb{H}##, the ring of real Hamiltonian quaternions. Define a function $$N:S\to\mathbb{Z}...

Hi, my name is alexander, i am student from Institute of radioprotection and dosimetry (IRD). My project is calculate MGD (mean glandular dose) from womans with augmented breast. i am having dificulties to calculate Kerma in air with mcnpx. I drew a block of air above the breast, i am using the...

Without a calculator, find all solutions w between 0 and 360, inclusive, providing diagrams that support your results. 1) cosW=sin20 2) sinW=cos(-10) 3) sinW< 0.5 4) 1<tanW

If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?

1 Is Equilibrium a Scalar or Vector quantity? 2 What is the unit of Equilibrium? Thanks & Regards, Prashant S Akerkar

Homework Statement Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector. Homework Equations d/dx(cosx)=-sinx The Attempt at a Solution I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...

I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...

When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead? Thanks

Homework Statement Givens: An object with a mass of 2kg has a momentum of p=<-1,-2,3>. The first two questions asked for the magnitude of the momentum and the corresponding unit vector, which i found to be 3.74 and <-0.267,-0.535,0.802> respectively. The next question asks for the speed of the...

Hello How to calculate electrical unit of Hydro power dam in rating 33.75*10^6 W What will be the energy Whr please help

Hello! (Wave) I want to solve the Laplace equation on the unit disk, with boundary data $u(\theta)=\cos{\theta}$ on the unit circle $\{ r=1, 0 \leq \theta<2 \pi\}$. I also want to prove that little oscillations of the above boundary data give little oscillations of the corresponding solution of...

Pick a random set of N points from the unit disc. Calculate the distance between all pairs of points and call the smallest value r. Do this calculation for many such sets. Please give me a hint how to estimate what the average value of r is. I guess a computer program could quickly come up with...

Is 'AMU' the unit for both 'atomic mass' and 'molecular mass'?

Hi I hope I'm not posting under the wrong forum but I'm pretty sure this is physics related. If you go to have an X-ray, in the film there will be information detailing the dosage you just received. However I encountered a rather unusual one like this: 70kV 100mAs 100mS -- dGycm2 El_s:300 As...

Homework Statement Hi there, I've been trying to gain some intuition on how the convolution sum works, but as I dig deeper I am realizing that there is an issue with my intuition of signals and systems, in particular the unit impulse response. My issue is trying to understand how a unit...

Dear All, I've been recently reading the very clear text of Burns and Glazer entitled Space Groups for Solid State Scientists in the context of my thesis which requires understanding of symmetries of crystals, more specifically symmetries of (approximate triply periodic minimal surfaces)...

Homework Statement Homework EquationsThe Attempt at a Solution I know the units for V is J/C so #3 and #4 are wrong. But how do I determine the signs?

Hello, I've been told that 'natural units' ensure ##\hbar=G=c=\mu_0=1## but... when I look for it in Wikipedia I find that there are (mainly) two kinds of natural/geometrized units: The Planck ones and the Stoney ones, (https://en.wikipedia.org/wiki/Geometrized_unit_system) Also, some...

Homework Statement Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$ Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0) Homework Equations $$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$ $$ z = \sqrt{x^{2}+y^{2} -1} $$ $$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...

Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...

What's the difference between zone sizing data and terminal unit sizing data in HAP and is it possible that the maximum cooling sensible load for the zone be greater than the Sensible coil load of the terminal unit as indicated in the system design report attached .Note that it is a single space...

i have been learning about using redshift to measure the distance of far-flung objects, but in all my research i never saw a specific unit used to quantify how redshifted an object is. to me it seems like an oversight, because such a unit would be very useful for the study of distant galaxies...

Hi, I need to convert a value of 12*10-16 m7 N-3 y-1 to the units of kPa-3 m2 s-1. Can anyone help me? where: m = meters, N = Newton, y = year, s = seconds. Thanks!

Hi all, I'm a bit undecided what to pick as my last unit for my math major. I'm about to the the 3rd year of my degree. I'm a physics and math major, although I'll be continuing with physics into honours (and beyond). I'm all set with my physics units. There isn't much leeway in accelerating...

Homework Statement A 0.5 kg block of ice is sliding by you on a very slippery floor at 2.5 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0035 seconds. The block eventually slides at an angle of 24 degrees from its original...

Hello, I have been looking to build a pmu similar to this device, it will be for domestic use. https://www.digikey.com/en/articles/techzone/2014/jan/low-cost-microcontroller-based-phasor-measurement-units-improve-smart-grid-reliability I am having a bit of trouble with the sampling part and...

Homework Statement A hydrocooling unit can cool fruits and vegetables from 30 to 5 C at a rate of 20,000 kg/h under the following conditions: the channel is 3 m wide by 90 cm high. The water is circulated and cooled by the evaporator section of a refrigeration system. The refrigerant...

Why radians are usually neglected in dimensional analysis?

Hi, Everyone, Hope you are fine and doing great. I am new here and need to ask how can we match both sides of F-N tunneling equation in unit. I am attaching the equation, i am working on but i could not solve the unit match problem. if the left term has unit A/V2 then how can we solve 2nd...