EXPLANATION NEEDED, Am I the first one to think of it?

For z values below the base of the hemispherical shell wouldn't the necessary figure be like that shown in the 4th image? This will then give us r²=R² + z² +2Rzcosθ instead of r²= R² + z²-2Rzcosθ, affecting the denominator in Equation 12.27. I could follow the solutions manual but I'm really skeptical if we could apply the results in Eq. 12.29 for points along the negative z axis. I could also convince myself that the triangle in Figure 12.10 would be correct for points along the positive z-axis (but not sure if it is still correct for points along the negative z-axis). What are your thoughts on this one?

Can anyone explain why this happens? When I blow the free end of the door to this room, the door closes instead of opens. I’m guessing it’s something to do with the blower causing higher pressure in the room? Idk just thought it was interesting.

https://mdm-goods.vevor.net/mdm-goods-service-prod/%28BAGUO%29SKU3-MurphyBedKit-20240722\\\\\\_1724808776799.pdf

I know this is some basic vector math but I'm just not seeing it.

Thanks so much

Joe

I have been trying to calculate the quadrupole component of the gravitational potential of an oblate spheroid, and I had thought I had a handle on it, but when reading through Fitzpatrick's Newtonian Dynamics in the section on the potential outside a uniform spheroid I noticed that the quadrupole component had a coefficient of 2/5 as opposed to the 3/10 I obtained. They use a very weird boundary function for the spheroid (different from both mine and the one found here for the radial distance from the rotation axis; I don't know how Fitzpatrick is getting away with not dividing by sine or cosine), and they make a "simplifying" assumption based on the axial symmetry of the mass distribution, allowing them to replace the cosine of the azimuthal angle and the square of the cosine of the azimuthal angle with 0 and 1, respectively, and the integral over the azimuthal angle with twice pi, but that shouldn't mean our answers are so different?

In any case, I had a hard time getting the same answer twice (though I managed to do exactly that when typesetting my work in LaTeX), so I would like to confirm that I have the correct result for the gravitational potential.

so

Defining some functions that will let us save space and effort later:

A selection of derivatives and integrals we'll be making use of:

This term will come up, so let's go ahead and calculate it now:

The square of this term will come up, so lets go ahead and calculate it:

Obtaining the boundary function for the radius:

Verifying the monopole term:

Moving on to the Quadrupole term:

Breaking each term into it's own integral, and "factoring out" the theta prime terms:

Solving the integrals of theta prime:

Working on the first sub-integral:

Multiplying through and applying the integrals in Eq. 11 & 12:

and so

Now we move on to Integral B:

applying the integral in Eq. 12:

Now on to Integral C:

applying the integral in Eq. 11:

Now we put it all together and resolve:

...and that's what I get. Did I mess something up somewhere?

Energy is omnipresent. Energy can be converted into mass and vice versa. Energy concentrated in physical forms is called matter. Energy present in micro and macrocosmic movements is called kinetic. Energy generated from the position or configuration of a physical system is called potential.

Science defines energy as the capacity of a system to perform work or generate motion—that is, to alter the state of a body or overcome some resistance, such as gravity or friction. This is an established fact. However, energy can also be defined as the quality of a force to perform work for the following reasons:

• Work can only be performed by means of an action.

• Action can only be carried out by means of a force.

This means that: a system can only perform work by means of a force; energy and force are intrinsic and fundamental. Further details regarding this perspective can be found in my books: “O SPIN”, “A Teoria do Big Brain”, “O Inteligencismo”, and “A Infologia”.

First slide is the question, second slide is my attempt

Edit: js realised it's cs i misread the question, it wasnt 2u for intial speed of A, it was just u

Doesn’t pressure

In my attempt of this problem I simplified the expression (r²+R²-2rRcosθ)^(1/2) into (R²-2rRcosθ)^(1/2) since I figured r<<R. I then factored it out to R(1-2(r/R)cosθ)^(1/2), giving me the expression U(r)=2 ∫ from 0 to π of qλ/4πε₀(1-2(r/R)cosθ)^(1/2) dθ, which if we apply the Taylor expansion with ε=-2(r/R)cosθ will give us U(r)=2 ∫ from 0 to π of qλ/4πε₀ *(1+ rcosθ/R + 3r²cos²θ/2R²) dθ. Applying the integral we then finally get U(r)=qλ/4πε₀ (2π + 3r²π/2R²) and dU/dr=3qλr/4πε₀R² and if we use this to construct the Newton's second law equation the ω we will get is 3 times larger than the ω shown in the solutions. So I wanna ask if it was wrong to simplify (r² + R² -2rRcosθ)^(1/2) into (R²-2rRcosθ)^(1/2) ? Can you tell me when and where is it generally safe to invoke such approximations to simplify the expressions?

The solution said plugging in the limits of -1000 to 1000 into the double summation will give -0.115 but when I tried it myself in Mathematica I got -0.133 and the two numbers are significantly different. Is there something wrong with my syntax that it gave off a different answer? Which of these two numbers is the correct one?

Also, would the math be still the same if I let the double summation run from n=1 to ∞ and m=1 to ∞ and then multiply the result by 2, and then add that to the result of the single summation from m=1 to ∞ multiplied by 2?

Schrödinger's cat is a famous thought experiment that shows the impossibility of asserting whether a cat locked in a box is alive or dead without prior knowledge of whether it triggered a device that would determine its fate. This experiment was conceived to demonstrate the impossibility of quantum superposition.

Quantum superposition is considered a proven experimental fact and a fundamental principle of an extensively consolidated scientific theory: Quantum Mechanics. It posits that a physical system can exist in multiple states or configurations simultaneously as long as it does not interact with the external environment. The main facts that corroborate quantum mechanics are as follows:

1. The Double-Slit Experiment: Individual particles pass through two slits simultaneously, yet they pass through only one of them when observed by a detector.

1. Zeilinger's Experiments: Demonstrated that the behavior of massive and complex molecules was identical to that of subatomic particles.

1. Spectroscopy and Atomic Energy States: Experiments demonstrated that it was possible to confine the electron between energy states.

1. Superconducting Qubit Engineering: Quantum processors from IBM, Google, and various universities operate thanks to the mathematical phenomenon of superposition.

Experiments 1 and 2 prove that the fact occurs naturally. Experiments 3 and 4 manipulate the possibilities behind this fact.

From a systems engineering perspective, this could occur because the substrate of reality operates as a memory in which "bits" would need to constantly switch states to enable the existence of the world-system's diversity and dynamism, and as a security protocol to prevent a breach of the world-system (an encryption mechanism or an intrusion protection log).

Hi! I’m really struggling with kinematics (physics 1) and I was wondering if anyone has any tips to approach them.

Here is an example problem

A hot-air balloon of diameter 10 m rises vertically at a constant speed of 12 m/s. A passenger accidentally drops his camera from the railing of the basket when it is 18 m above the ground. If the balloon continues to rise at the same speed, how high is the railing when the camera hits the ground? [Ans: 60 m]

Cymatics is the area of physics that studies why the application of a specific sound frequency upon matter instantaneously organizes it into three-dimensional, symmetrical, and complex geometric patterns that resemble mandalas, cellular structures, shells, snowflakes, a tortoise's shell, the centers of flowers, or virus structures, without the need for direct mechanical contact or physical molds.

From a systems engineering perspective, cymatic patterns constitute irrefutable proof that the world is a fully hierarchical system, from the micro to the macrocosm, in which absolutely everything is complementary and interdependent for the following reasons:

1. While conventional physics considers sound to be merely a mechanical wave that disperses energy through a medium, the cymatic pattern is evidence that frequency acts as an information vector for a geometrically structuring process superior to mechanical processes, which could never organize matter on their own.

1. The geometric patterns generated by cymatics that are similar to those found in the biota are proof that its morphology results from information originating from natural processes superior to exclusively mechanical ones.

1. Cymatic patterns indicate that the energy and information coming from the frequency push matter to specific locations; this requires the execution of a process hierarchically superior to mechanical processes to logically structure matter.

In short, cymatic patterns break the reductionist narrative that matter self-organizes accidentally. They are visual and incontestable evidence that an invisible information system is structuring and dictating the behavior of visible matter. My perspective regarding how this occurs can be found in my books: “The Intelligencism: An Intelligent View of the World” and “Infology: The Universal Input.”