It is said that value of $g=0$ at the centre of the Earth. $g$ is calculated using the formula                                      $$   g=G*M/R^2.$$  When $R = 0$, $g$ approaches infinity. I have two questions.   Why is the $g= G*M/R^2$ not working? How to calculate the value of $g$ at centre of earth?
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Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
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My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :)
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I understand GCD mathematically but i can't figure out where to apply it. For eg I saw this problem today:     Adam is standing at point $(a,b)\in\mathbb Z^2$ in an infinite 2D grid.   He wants to know if he can reach point $(x,y)$ or not.   The only operations he can do is to move to the points $(a+b,b), (a,a+b), (a-b,b)\text{ or }(a,a-b)$ from some point $(a,b)$.   It is given that he can move to any point on this 2D grid, i.e. the pointscan have positive or negative $x$(or $y$) coordinates.      Tell Adam whether he can reach $(x,y)$ or not.   The solution to this problem is simple: If $\gcd(x,y) = \gcd (a,b)$ then adam can reach $(x,y)$ from $(a,b)$, otherwise not.    While this solution is indeed working I am not able to figure out how to come up with it. What is the significance of $\gcd$ in this question?
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Can someone explain to me, step by step, how to calculate all infinite values of, say,   $(1+i)^{3+4i}$?   I know how to calculate the principal value, but not how to get all infinite values...and I'm not sure how to insert the portion that gives me the other infinity values.
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Edit : Added a picture for better understanding of my querry. All the texts I have studied has used an non-inertial frame to explain the phenomenon. But every time I see something explained with pseudo forces, I try to realize in real forces.   But in this case I tried to explain it in a frame which is centered in Earth's center and not rotating. But I couldn't explain what happens to the tangential component (to the point on Earth's surface where $g$ to be measured) of centripetal acceleration in this scenario. It doesn't cancel out.   But then I thought the tangential component is so low that practically it would have no effect.  Is my explanation wrong?
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Special mention to the  that kicked this off:  Anyway, the host points out that a tunnel that connects a pair of points on the earth's surface can be thought of as a gravity train - where the force of gravity along the tunnel allows an object to fall through it and emerge on the other side. Any force perpendicular to the tunnel (in case the tunnel doesn't pass through the center of the earth) is ignored.   I worked it out and the equation is $a = -\frac{4}{3}\pi\rho G d$, where $d$ is the distance from the center of the tunnel and $\rho$ is the density of the earth.   Clearly, it is simple harmonic and therefore the period is constant and has no dependence on which two points were used to make the tunnel.   Does anyone have an intuition for why this should be the case? I imagine some Gauss' law type of argument should work here, but I cannot see it.  EDIT: More to the point, why even when the tunnel does not pass through the center, one obtains the same period. Is there any deeper explanation as to why this should be the case?
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Let us say a block of mass is placed on the surface of earth. Then while drawing the forces on that body, we say:   Force $F = mg$ acting towards the center of Earth. Normal reaction $N$ offered by the surface of Earth.   If no other forces are acting, we say $F=N$. But what about the centrifugal force $m\omega^2R$ . Why don't we ever bring that into picture? What am I missing?
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I have tried a little bit to solve the problem which goes as follows:   My intuition says that there exist no $f:\Bbb R\to\Bbb R$ such that $$f(g(x))=x^{2018}\text{ and }g(f(x))=x^{2019}.$$  Note that $$f(g(f(x)))=f(x)^{2018}\implies f(x^{2019})=f(x)^{2018}$$  Similarly, $$g(x^{2018})=g(x)^{2019}$$ Putting $x=1$ in $f(x^{2019})=f(x)^{2018}$, we get $f(1)=f(1)^{2018}$ and thus $f(1)=0$ or $f(1)=1$.  Similarly, putting $x=1$ in $g(x^{2018})=g(x)^{2019}$ we get $g(1)=g(1)^{2019}$ and thus $g(1)=0,1,-1$.  Now, I can't proceed further. Can anybody solve it? Thanks for assistance in advance.
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I'm sorry if the answer is obvious for you guys, but why don't we all (including buildings, road, people, the ground) collapse to the center of the earth because of gravity? Is it because we have velocity, just like the earth not falling to the sun (or electrons not falling to the protons)? But that analogy doesn't sound right because those involve wide empty space.
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The idea that as you move through the earth you get a symmetric cancelling of gravitational acceleration which approaches zero acceleration due to gravity at the center of the earth makes a lot of sense I think. I also think that is a pretty easy idea to understand physically.  On the other hand, I am just beginning to study and understand the general relativistic explanation for gravity and I'm wondering how this same idea is explained using general relativity. That is, what is the explanation for why gravitational acceleration decreases (linearly if I'm not mistaken) as you approach the center of the earth using general relativity? Specifically, I would like to understand what is going on with the curvature of spacetime in this scenario? Is it becoming less curved? Some math would be appreciated in answers!
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The Problem For a small mass a distance $R_i$ away from the center of the Earth, how long would it take for the object to fall to the surface of the Earth, assuming that the only force acting upon the object is the Earth's gravitational force?  Relevant Information The following discussion seems to have solved exactly the same problem:   However, upon working out the mathematics, I'm not exactly sure how to evaluate the constant of integration.   A Partial Solution $$ F=\frac {-GMm}{s^2} $$ $$ a=\frac {-GM}{s^2} $$ $$ \frac {dv}{dt} = \frac {-GM}{s^2} $$ Multiplying by $v$ and then integrating by $dt$ on both sides, we have $$\frac {1}{2} v^2=\frac {GM}{s} +c_1$$ where $c_1$ is a constant of integration. Substituting initial conditions of $v=0, s=R$, we have $$\frac {1}{2} v^2=GM(\frac {1}{s}-\frac{1}{R})$$  At this point of time, when I use Wolfram Alpha, I get  $$c_2+\sqrt{\frac{2}{R}}t=\frac{\sqrt{s}(s-R)+R\sqrt{R-s}\times{\tan^{-1}(\sqrt{\frac{s}{R-s}})}}{\sqrt{GM(R-s)}}$$  where $c_2$ is a constant of integration. Substituting initial conditions of s=R, t=0, we find that the term $$\tan^{-1}(\sqrt{\frac{s}{R-s}})$$ is undefined. At this point, I'm stuck. Any ideas on where I've made the mistake here?  (For those interested, this question was inspired by the Greek myth which states that a bronze hammer dropped from heaven would take 9 days to hit the Earth and would reach on the tenth).
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What are the exact values of $\cos(2\pi/7)$ and $\sin(2\pi/7)$ and how do I work it out?  I know that $\cos(2\pi/7)$ and $\sin(2\pi/7)$ are the real and imaginary parts of $e^{2\pi i/7}$ but I am not sure if that helps me...
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For whatever reason, I seem to have gotten myself twisted into quite a confusion on how to process data from my 3 axis gyro and 3 axis accelerometer to get gyro rotation values related to the earth's reference frame. i.e. Given my accelerometer measures gravity (we assume it's stable) along the sensors (x,y,z) reference frame, how do I adjust the values of my gyro, measured along the sensors same (x,y,z) reference frame such that I am actually measuring the rotating about the Earth's frame x (North), y (West), and z (downwards) direction?  I just can't seem to wrap my head around the use of the rotation matrix. Do I simply employ a 3d rotation matrix, and multiply the vector, \$[gyro_x, gyro_y, gyro_z]^T\$, by a rotation matrix, R? I then assume the rotation angles, are somehow derived from the filtered accelerometer outputs, but I'm having trouble deciding on what those should be.  I have yet to find a concise reference on how to move the rotation and acceleration values from the sensor reference frame into the earth global reference frame. Perhaps someone can help? Thanks
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When I was in the university (in the late 90s, circa 1995) I was told there had been research investigating the $2$ (the square of distance) in the Newton's law of universal gravitation.  $$F=G\frac{m_1m_2}{r^2}.$$  Maybe a model like   $$F=G\frac{m_1m_2}{r^a}$$  with $a$ slightly different from $2$, let say $1.999$ or $2.001$, fits some experimental data better?  Is that really true? Or did I misunderstand something?
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I have been trying to imagine ways to visualise the gravitational field of the earth... It's near zero at the outer edges of the atmosphere, strong at sea level, strongest at the quarter point, then drops off to zero again towards the middle as the masses cancel each other out... Something like a 3D donut?  This got me thinking about black holes. Would they also have an area of zero g right at the centre where all the surrounding mass cancels each other out?
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Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$? I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it should commute with every element of $R(G)$. In particular it should commute with $g=1 \cdot g \in R(G)$ Thus we get $$\big(\sum_{i=1}^{n} r_i g_i\big)g=g\big(\sum_{i=1}^{n} r_i g_i\big)$$ Thus I get $$g_i \cdot g=g_i \cdot g \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \forall g \in G$$ Hence $x \in Z(R(G)) $. Is this correct?
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Calculate the potential and the intensity of the gravitational field at a distance $x&gt; 0$ in the axis of thin homogeneous circular plate of radius $a$ and mass $M$.    Could anybody describe how to calculate this? Slowly and in detail. I'm helpless.  Answer is: potencial $\phi = - \frac {2 \kappa M}{a^2}(\sqrt{a^2+x^2}-x)$ and intesity $K = \frac {2 \kappa M}{a^2} \left( \frac{x}{\sqrt{a^2+x^2}}-1 \right)$