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The heat conduction equation for a semi-infinite slab with a boundary condition of the first kind is as follows: The problem is delta is a very small number, so the first exponential will tend to infinity. I am programming this in Fortran and it can accommodate values up to magnitude of 310...

Ok, I have actually found the answer from http://www.bristol.ac.uk/phys-pharm-neuro/media/plangton/ugteach/ugindex/m1_index/med_memb/file/Nernst1.htm. Basically, a convenient way to analyze these equations is to take the log of both sides. Since e takes the natural log and the equations are...

There we go! This is what I was looking for. Work integral is path dependent. Thanks!

The total work is (myforce - friction) x 5 + (myforce - friction) x 5

I am going to stop pushing on it so the friction dissipates the energy and it stops. Then I move to the other side and push the object back.

Ok, I still don't know why you are asking about friction since friction would just be calculated into the net force. But the friction's work would be whatever the friction force is times the total distance (not displacement since friction is not conservative), which will be less than the pushing...

Stopped by friction. As in if I stop pushing then it doesn't move. And friction points in the opposite direction. But friction is a dissipative force

I know that by physics definition if displacement is zero, work is zero. However, if I push an object 5 m to the east, and then move to the other side of the object and push it 5 m back to the west. I think in this case I have always done positive work on the object and hence the total work...

Hello all, This is knowledge needed to solve my take-home final exam but I just want to ask about the definition of Bessel's integrals. This is not a problem on the exam. Wikipedia says the integral is defined as: $$J_n(x) = \frac {1} {2\pi} \int_{-\pi}^{\pi} e^{i(xsin(\theta) - n\theta)} \...

It is exactly as written in the book. Actually, I just figured it out... In an orbit, the r is defined from the center of the inertial frame. So r2 is rotated from r1 but the root of the vector is still at the inertial center. That means delta_r is a tangential vector that connects r1's vector...

Hello all, I have a question regarding the precise definition of angular momentum in orbital motion. I see one definition says angular momentum h, position, r, and radial velocity, r_dot, are related as follows: h = r x r_dot. However, I also see one definition that says h is related to r and...

Additionally, is the following relationship between circular tangential velocity, Vt, orbital speed, Vo, radial speed, r_dot, and flight path angle, Φ, correct? Vt2 + r_dot2 = Vo2 Therefore, Φ = tan-1(r_dot/Vt)

In the case of a satellite orbiting the Earth. Would the relative velocity calculated by the Vis Viva equation the same as the orbital speed of the satellite? If the satellite is in an ellipse, would Vis Viva give the circular tangential velocity Vt=rω, or would it give the orbital velocity...

I tried. I am looking for the specific algorithm that finds the unique grid points, and searching for Triangle Numbers gives too many mathematical theories. I will try Gauss sum when I get back. Thanks

I was investigating the number of unique grid points in a Cartesian coordinate system if I were to start at a corner (say coordinate 1,1,1), and make one step in each of the three positive directions (coordinates 1,2,1; 2,1,1; and 1,1,2). Now I went from 1 point to 3 points. I repeat the same...

Alright, thanks! I thought kWh is basically kJ. I guess kWh needs to convert by a factor of 3600 before it becomes kJ then.

Say I have a 60W device. I know running for one hour it uses 60 W x 3600 s = 216 kJ of energy. Does that mean it uses 216 kWh of energy? Or does kWh means 60 W x 1 hour = 60 kWh? I am confused because the units in 60 kWh is not in their simplified terms. More specifically, this pertains to...

Hello all, this is related to a project that I am working on. It is not directly related to the project but as part of it, I thought it would be a good idea to check the temperature difference that I need to maintain in order to effectively transfer a certain amount of heat between the TH in a...

If we are going to investigate an ideal case, then there are assumptions you can make to have the Rankine cycle approach Carnot efficiency. Such assumptions may include: the turbine exit quality can be neglected, the pump may pump a mixture, and the components in the cycle does not generate...

I personally preferred energy methods to relate speed, height, and other forms of energy devices such as springs, frictional surfaces, etc. It is simply another way to solve kinematic problems that many people (including myself) find easier and more intuitive to use than regular kinematic...

This question is related to a project. I have the mass flow rate, RPM, and all other related information regarding a refrigeration cycle, does anyone have an idea on how can I estimate the total amount of working fluid I need to put into the device? In case this comes up: the device is going...

The problem statement says the pressure in the wind tunnel can be changed to compensate for the limited speed from the fans, but it doesn't say how does it change the pressure. Depending on different ways to change pressure, the equation is different. If I assume the normal stuff like...

I need 250 m/s of airflow speed in a wind tunnel. I can assume for this purpose, the test section of the wind tunnel has uniform flow. However, the wind tunnel can only generate a flow rate of 100 m/s but the pressure can be increased. I am drawing a blank on how could a generic boundary layer...

Yes I did. The equation maker here is very bad with exponents, got me all messed up.

Hmm if the fundamental theorem is, quoting you, "every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors", then I do not believe I was using it. I was simply deducing that since p^2 is perfect...

The image is extremely awkwardly worded. You should not feel bad for not getting it. It is just a terrible way of teaching this topic. What happened is that it recognized that ##1/x## can be re-written as x^-1, and then you apply the power rule of differentiation: derivative of ##x^n## is...

What you have said is very interesting, I have never thought of it like this before. To me, the equation you have ##3+4-2\times(2+4)##, is reduced to a + b - c, where c = ##2\times(2+4)##, and since c = ##2\times(2+4)##, I will now compute c first. I see c as: a x b, where a = 2, b = 2 + 4, so...

But just to put it in variables. if square root m = r = p/q. Then m = r^2 = p^2/q^2 Then m*q^2 = p^2, and that means p^2 has m as one of the factors. Since p^2 is a perfect square, there must be another m as a factor, and that means p also has m as a factor. Now q^2 = p^2/m. Since we have...

I dare say that the same process can be used for square root of 3. Let square root 3 = p/q. 3 = p^2/q^2 p^2 = 3q^2, showing p^2 has factor of 3, since it is squared, 3 can't stand by itself, there must be another 3, indicating that p also has a factor of 3. Now q^2 = p^2/3, since we have...

Actually reading your post in full detail and the image from the book of square root of m = r = p/q, that is exactly what I proved in my above reply.

You can prove it by proving square root of 2 is irrational. The ancient greeks thought that all numbers can be represented by a fraction of two integers. So square root of 2 should be able to be expressed as p/q, given p and q are integers prime to each other (no common factors). Now if we...

That makes sense. Thank you.

Sorry I meant to say the half sphere facing us because clearly we don't see behind a sphere. And from the half sphere we can still see top of the sun and especially any solar flares from it. But I am wondering that we shouldn't be able to even see that half entirely since only a small portion of...

What I am trying to ask is that only such small portion of sunlight hits the Earth, we shouldn't even be able to see half of the sun facing us. We should only be able to see a small circular area on the Sun's surface

Exactly because light is emitted from all directions, only a small part reaches Earth. If only such a small portion reaches earth, what we see through our telescopes should only be that small portion of the sun in the sky. However, when observed in the telescope, we can see the entire Sun such...

Not quite what I am asking...

According to calculations, only 0.000000724654% of sunlight reach earth, and if we can only see an object if the light bounced off the object hits our retina in the eye, then how can we see the entire Sun through any means (telescope, etc.)?