Physics, 09.10.2019 05:00 powellalyvia

Ageostationary satellite is a satellite which always hangs above the same location on the earth. that is, as the earth spins under the satellite once every 24 hours (actually 23 h 56 m 4.1 s, you can look through the wikipedia article on "siderial time" if you want to know why it is not exactly 24 h), the satellite completes exacly one orbit. find the radius of a geostationary orbit. the mass of the earth is 5.97 × 1024 kg. explain your reasoning!

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Answer from: nathansnyder200

r=42227Km using 24h, r=42150Km using the exact given value.

Explanation:

The force that acts on the satellite of mass m is the gravitational pull of the Earth, of mass M. If the distance between their centers is r, we know that this gravitational force must be:

$F_G=\frac{GMm}{r^2}$

Where $G=6.67\times10^{-11}m^3/Kgs^2$ is the gravitational constant.

The satellite moves in a circular trajectory because the net forces acting on it are centripetal, so we write the equation of the centripetal force:

$F_{cp}=\frac{mv^2}{r}$

Since only the gravitational force is acting on the satellite this force is the net force, and thus, equal to the centripetal force:

$F_G=F_{cp}$

Which means:

$\frac{GMm}{r^2}=\frac{mv^2}{r}$

Or:

$\frac{GM}{r}=v^2$

The velocity of the satellite is v=C/t , where C is the circumference of the orbit, whose radius is obviously r: $C=2\pi r$ , so we can write:

$\frac{GM}{r}=(\frac{2\pi r}{t})^2=\frac{4\pi^2 r^2}{t^2}$

Which means:

$r^3=\frac{GMt^2}{4\pi^2}$

Which is Kepler's 3rd Law for a circular motion. We can write this as:

$r=\sqrt[3]{\frac{GMt^2}{4\pi^2}}$

Since there are 60 seconds in a minute and 60 minutes in an hour, using 24 hours we have:

$r=\sqrt[3]{\frac{(6.67\times10^{-11}m^3/Kgs^2)(5.97\times10^{24})(24\times60\times60s)^2}{4\pi^2}}=42226910m=42227Km$

We could use the exact time of (23)(60)(60)+(56)(60)+(4.1) seconds, and in that case we would obtain r=42150Km

Answer from: Quest

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

$x(t)=ax(x-6)$

where $a$ is unknown. because the position is parabolic, there is symmetry about the midpoint at $t=3$ , and we know that

$x(3)=10\implies -9a=10\implies a=-\dfrac{10}9$

so that the position function is

$x(t)=-\dfrac{10}9x(x-6)$

which means the velocity is

$v(t)=\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{20}9x+\dfrac{60}9$

then the total distance traveled is

$\displaystyle\int_{t=0}^{t=6}|v(t)|\,\mathrm dt=\left\{\int_{t-0}^{t=3}-\int_{t=3}^{t=6}\right\}v(t)\,\mathrm dt=20$

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at $x=0$ and ends at $x=0$ , so its displacement is 0.

Answer from: Quest

The correct choice is c

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Ageostationary satellite is a satellite which always hangs above the same location on the earth. tha...