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[3] Motion Under Force that is Function of Position. A particle of mass m [alls from rest toward sphere with much larger mass M (m < < M) due gravitational fo...

Question

[3] Motion Under Force that is Function of Position. A particle of mass m [alls from rest toward sphere with much larger mass M (m < < M) due gravitational force acting On it, as described and discussed on page 4 of the course notes to the (a) Starting with equation [15], derive equation [16] (see page 4 of the course notes) which relates the position of the particle to the time elapsed _ Show all of the steps Mid explain your work_ Do not use table of integrals: do the work yourself Show

[3] Motion Under Force that is Function of Position. A particle of mass m [alls from rest toward sphere with much larger mass M (m < < M) due gravitational force acting On it, as described and discussed on page 4 of the course notes to the (a) Starting with equation [15], derive equation [16] (see page 4 of the course notes) which relates the position of the particle to the time elapsed _ Show all of the steps Mid explain your work_ Do not use table of integrals: do the work yourself Show that the equation gives t =t0 (b) Take ro IORE; where Rg is the radius of the earth. Neglecting air resistance and all othe gravitational forces, how long does take the particle to reach the surface of the earth? (c) Consider the case where both masses I audl M are point Hasses particles_ What is the tin taken for m to reach M? Finally; by thinking very carefully, and drawing On your knowled the motion of comets, show that the result for T' is in agreement with Kepler's third law. Vir) = %= fGm ($ ~%) ['s] J6/n #+Etrs) - Wtlsn J 7o t ["] r



Answers

Gauss' Law for gravitation The gravitational force due to a point mass $M$ at the origin is proportional to $\mathbf{F}=G M \mathbf{r} /|\mathbf{r}|^{3}$ where $\mathbf{r}=\langle x, y, z\rangle$ and $G$ is the gravitational constant. a. Show that the flux of the force field across a sphere of radius $a$ centered at the origin is $\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G M$ b. Let $S$ be the boundary of the region between two spheres centered at the origin of radius $a$ and $b$ with $a<b$. Use the Divergence Theorem to show that the net outward flux across $S$ is zero. c. Suppose there is a distribution of mass within a region $D$. Let $\rho(x, y, z)$ be the mass density (mass per unit volume). Interpret the statement that \[ \iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi G \iiint_{D} \rho(x, y, z) d V \] d. Assuming $\mathbf{F}$ satisfies the conditions of the Divergence Theorem, conclude from part (c) that $\nabla \cdot \mathbf{F}=4 \pi G \rho$ e. Because the gravitational force is conservative, it has a potential function $\varphi .$ From part (d) conclude that $\nabla^{2} \varphi=4 \pi G \rho$

All right today we're solving this interesting thought experiment where if you have a hole that goes straight through the earth from Point A all the way to the opposite side, through the diameter through the center um You know if you drop a mass M what would it do? So the first question we've got we've got a radius X. Of some point throughout the middle of the earth and the full radius of the earth are calculate the mass of that smaller sphere if they are concentric circles. So we're going to use to relations we're going to use the fact that the density of the earth is the same throughout the density of the small, spherical is the density of the full earth and density is equal to mass over volume. So we've got the mass of that little small sphere divided by the volume of a sphere which is four thirds pi x cubed. Where X is the radius of the smaller sphere is equal to the mass of the entire earth. Capital M divided by four thirds pi capital r squared. Now we see we have four thirds pi on each side, we divide them out. If we saw for the little mass M, we get little M is equal to big M times the ratio X over are cute from there. We can solve for the force of gravity using our classic Newton's equation of universal gravity. So we've got the gravity constant G times the little mass, M times the bigger mass, which is the still the small concentric circle with a dime or radius of X divided by the distance we are, which is X squared. Now we can sub in capital M. With our expression from part A. So we get g little M mass of the Earth times X cubed divided by the X squared from before times where it is cubed. Now we can see we have X appearing twice. We can cancel it out and we get the force of gravity is equal to G. M. M. X. Over our cute remember Little M. Is the mass of the small object. Capital M. Is the mass of the earth. Now we're tasked with figuring out how this is simple harmonic motion. So if recall at the center, if you plug in X is equal to zero, you're going to get a force of zero. But you know that the kinetic energy will be at its max because it's sped up from the force felt all the way to point A If you look at pointe au of a maximum force but zero velocity as a result of the object will oscillate along that whole through the Earth passing through the center and then filling an opposite force until it reaches the opposite side and then it will go back and go back and forth. Now we're gonna go to part D. Were tasked with finding the period of that oscillation. We're going to take the same period equation from her spring mass damper so the period T. Is equal to two pi times the square root of mass over stiffness. K. Now we don't have a spring here. So it's kind of weird. We don't know what the stiffness would be if we can figure out an equivalent stiffness by looking at our force equation and part beat. So we know that F. Is equal to G. M. M. X. Over our well Gm argued Gmm over. R cubed is roughly Kay, if we look at Hook's law which is F. Is equal to K. X. So we get a stiffness of G M. M over R cubed. Thus we substitute these two into our period equation. We get to pie times. The mass is still M. To buy the by G M M. And then the rQ goes to the top. We can cancel little M. Thus we get a period of two pi radius cubed, gravitational constant G times mass of the Earth M. Now in part E were given numbers for all three of these in addition to the constant to pie and you can solve to get a period of 85 minutes Now, part F asks if there is a satellite that is orbiting the Earth at a radius of our, what would be the period here. Now, if you look in our period equation, everything is a constant about Earth radius R, mass M of the Earth and gravitational constant G. In addition to the two pi. So if it's orbiting around the Earth, you'll actually have The same period since it's a function of the same three constants. So that's the period is the same at 85 minutes, I don't know.

In this question we have a figure which is showing that a satellite is moving on a circular path and the mosque of the satellite is small um in the center of the path we have a planet. All the math is capital. M the orbit radius is are not and the spirit of the satellite is we we are required to prove peter is calls to VT upon are not and the position vector party recalls to are not co sign VT upon are not coma are not fine VT upon are not. Then we need to find the satellite's exploration after that. We need to prove we square articles to GM upon are not and then we need to prove we into teeth calls to to buy are not and finally we are required to prove these particles to four pi Squire divided by Gm into are not cube. So let's see how to solve this question. From the figure we can observe that the ark land from The vehicles to 0 to the existing position of em will be arc land is equals two are not into data and this article and will be called to we into T. So from here we can right angle theta equals two. Bt divided by are not we know that the position vector party can grittiness are not because I intend to coma are not scientific to no substitute the value of data in this expression so we get the position vector parties equals two are not frozen VT upon are not coma are not sign VT upon are not So these are the file answers for part A and you know let's move to part B. Mhm We have obtained the position vector and we know that the formula to calculate. We're also director vedic and readiness. Day by day. T of position vector. Artie north of it to the value of position vector. So we get velocity vector Beauty. The calls to D. By duty are not person wait upon are not comma are not sign VT upon are not so this will be calls to Dubai Duty of are not cosign we T upon are not coma. Dubai Duty of are not sign VT upon are not no differentiate these functions so we get velocity that Beauty. The calls to minus V. Sign VT upon are not comma. We go sign we T upon are not so in this manner we have calculated that the director and now The formula to calculate exploration vector 80 can brightness D by duty of velocity vector Beauty now substitute the value of velocity vector so we get acceleration vector. Eighties calls to Dubai Duty of minus V. Sign VT upon are not comma. We cosign VT upon are not and this will be calls to Day by duty of -3. Sign Beauty upon are not coma. Dubai Duty of we co sign Beauty upon. Are not now to finish your these functions so we get minus we square up on are not cosign VT upon are not coma minus. We square upon are not fine VT upon are not And from here we can take out minus we square upon are not as a common. And this will be multiplied by cosign We t upon are not coma sign VT upon are not. So this is the exploration of the satellite and the final answer for part B. And now let's move to part C. Yeah, from the expression of exploration, we can observe that if we multiply and divide this expression by are not So we can write exploration like -3. Carbonara esquire are not cuisine VT upon are not comma are not sign VT upon are not and we have already obtained The position vector which is equals two. Are not chosen beauty upon. Arnold Coma are not signed beauty upon Arnold. Hence exploration, it will be called to minus we square upon our notice. Choir position vector already. No apply Newton's second law of motion. So we know that 4th f equals two. Moss into exploration. Yeah. Hence yeah, fourth will be called to minus M into we buy are not to the power to uh And let's say this is a question one. No apply. Mhm. The law off gravitation. So we can write 4th f. is called stew minus G. M mm upon. Are not esquire multiplied by are upon. Are not here. G is the universal gravitational constant. And let's say this is equation two. No, it's where the question went into so we get minus G. M. M up on are not a Squire multiplied by higher by are not It equals two -M. We upon are not to the power to into are many for the calculated this finally we get we scared the culture too Gm upon are not so this is the final answer for part C. And you know let's move to party. We know that to complete a cycle. Satellite moves peter is calls to to buy meridians we have already compared to the value of chitin part A So now substitute teacher is called to VT upon are not and this will be close to so by and when you further calculate this finally we get We into T equals two Dubai are not so this is the final answer for party and you know finally we are small to party from the elbow expression what is considered to buy Arnold we can right vehicles to so if I are not divided by T and then I'll substitute this value in expression we square records too Gm divided by are not so This will be equals two to buy are not by T To the power to equals two Gm upon are not and This will be calls to four pi square. Our notice choir Divided by the squared equals two Gm A phone are not many for the calculated this we get one, appointee squared equals two Gm upon four pi square are not To the power three Therefore, we can conclude that please travel, be calls to four pi square are not cube divided by G into em. So this is the final answer for this problem. I hope you understand the solution. Thank you.

In this problem. I'm doing the Diagram 1st. Just look at it carefully after that. I will simplify the problem. So here's the value it uh here is the man standing there. This angle is pita not he had this side, the angle age it cannot. This is also our here I can write devaluate MG. Here I can ride evaluate bt to and here I can write evaluate and omega not so so I'll bring it further. I can write the value as let em omega notice pseudo force because observer at disappeared so I can write a question as and be not described by R is equal to MD because he cannot minus M. Omega not scientists are not also I can write, we notice squared is equal to R. G. Gosh tita not minus our omega not signed he to not let it be question number one, I can write the value of w all horses is equal to care minus scary W pseudo plus W M G is equal to Kiev minus scary on going forward. I can also write develop M omega not assign tita not plus M D. Ar minus are caused tita not which is equal to have and be not E squared. So I'll bring it further. We notice square is equal to omega not r. Sine theta not plus two. RT 1 -4 Free to not let it be question #2 on further simplification from the question want to I can write the value peanut is quite difficult to our minus B, not B squared by our plus two. Gr We're not described is equal to 2G. Al by three. So finally, I can write, you know, difficult to under route who? Oddity by three. So putting it in the question one. I could get the value of course, cheater not equal to two plus and route and the white plus, see and cube. Bye. Three one plus and the square weird and is equal to w not by sea.

To coordinate eyes. The orbiter we have R. Of T. two vehicle to our notes. Cause T. To I let's I don't know it's a sign. Sita peter G. So you have I hear right out. I bless that's and the distance the just dance shovelled along the circle along the circle in a sign. Seeing she is V. Same C. That is reads times time. This is the reeds times time which records wish equals the circular a claim. Oh no tita Which implies that. See it's uh it's equal to V. T. Divided by our notes. This implies that um So this implies that artsy. It's equal to uh No it's because theater is VT divided by our notes. I legs. Oh no it's sign. We C divided by our notes. G. That is swimming. When we coordinate size their orbits are clean. This is what we have. Then the second part is to be find the acceleration of. There's such lights. We have V. See the vehicle C. D. R. D. C. Which is equal to so have -5 sign. We C divided by our notes. I leads because we C divided by our notes steve. So this implies that e acceleration E is going to be devi over the sea. And this would give me minus V squared divided by Oh no, it's of course we t divided by our no, it's I plus the differential of that is we squared divided by our notes sign. So that is also minus here because if you do friendships close, yeah, getting my next sign. V C divided by our notes jean and this is equal soon minus 50 squared divided by our notes. I have our notes cause we C divided by our nuts. I less. I don't know it's sign. We C divided by our notes. She So this is the acceleration. So this is this will virtually give us mine it's we squared divided by I'm not squared are of C from that because you know what's Rfc is from here, R O C. So hey, this is the celebration of the such lights. Then the nest is to find the force which is equal to M E. And we want to prove that's so see see you have F to be equal to M mm. So this implies that I have flaws to have minus she mm hmm divided by I'm not squared. I have are divided by our notes to be called to em times A we know what A is. It's minus V squared divided by our north squared times. So what do we have here? So I have minus Gm divided by our cube are not cute. Am here councils that and this is going to be equal to My name's MV 2ared divided by our notes squeeze. So this implies that our V squared, it's going to be equal to she am divided by our notes. So what happened? He realized that this and that goods are we then these councils. Does she have one of the guys Sufi, It's going to give us she? M divided by their So then we have been able to show that this is equal to that's then there. Yes, but deal did I say? Let's see is the time is it time for? But there's such lights to complete one full of it's one food updates. So this implies that V. C. It's equal to the circumference of a psycho. So there's a good friends of a cycle is soup. I our notes bangs the last bus. If we substitutes, if we substitute V two vehicle 2 to 5, soup by our notes. Divided by C. See into the squared GM divided by our notes. This implies that we have so This implies we have four pi four pi squared are not squared divided by T squared to be equal to G M, divided by uh nice Sudan howard C. So this implies that r t squared is going to be go to for I four pi squared, I'm not squared are not skewed because this woman supply that so I'm not so cute divided by G I am. So what does this mean? This implies that our C squared is proportional. Yeah, is proportional. Super proportional is directly proportional. Two. Our notes cube says this whole thing is a constance. So since this, since four pi square divided by G M is a constance.


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