follow to textbook knowledge, the massive of the planet is about $6 × 10^24\,\mathrmkg$. How is this number determined when one cannot simply weight the planet using consistent scales?

According come Newton"s law of Gravity based on attractive force (gravitational force) that two masses exert on every other:

$$F=\fracGmMr^2$$

Where:

$F$ is the gravitational force$G = 6.67 \times 10^-11\ \mathrmm^3\ \mathrmkg^-1\ \mathrms^-2$ is a constant of proportionality$M$ and also $m$ are the 2 masses exerting the forces$r$ is the distance between the 2 centers the mass.

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From Newton"s 2nd law that motion:

$$F=ma$$

Where:

$F$ is the force applied to an object$m$ is the mass of the object$a$ is the acceleration as result of the force.

Equating both the equations:

$$F = \fracGmMr^2 = ma$$

$$\fracGMr^2= a$$ (The $m$"s canceled out.)

Now deal with for $M$, the fixed of the Earth.

$$M = \fracar^2G$$

Where $a = 9.8\ \mathrmm\ \mathrms^-2$, $r = 6.4 \times 10^6\ \mathrmm$, and $G = 6.67 \times 10^-11\ \mathrmm^3\ \mathrmkg^-1\ \mathrms^-2$.

$$M = 9.8 \times (6.4 \times 10^6)^2/(6.67 \times 10^-11)\ \mathrmkg$$

Hence,

## $M = 6.0 \times 10^24\ \mathrmkg$

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Note: i updated this prize to encompass a summary of the historical techniques.

Historical Techniques

Newton occurred his theory of gravitation mainly to define the motions of the body that kind the solar system. He additionally realized that while gravity provides the earth orbit the Sun and also the Moon orbit the Earth, that is also responsible for apples falling native trees. Whatever attracts everything else, gravitationally. That argued that one could in theory measure the gravitational attraction in between a pair of small spheres. Newton self realized this, however he didn"t think the was really practical. Absolutely not two little spheres (Newton 1846):

Whence a sphere of one foot in diameter, and also of a prefer nature to the earth, would entice a tiny body placed near its surface ar with a pressure 20000000 times much less than the planet would carry out if put near its surface; yet so tiny a force could produce no judicious effect. If two such spheres were distant but by 1 of one inch, they would not, also in spaces void the resistance, come with each other by the pressure of their shared attraction in much less than a month"s time; and less spheres will certainly come with each other at a rate yet slower, namely in the ratio of your diameters.

Maybe a mountain?

Nay, totality mountains will not be adequate to produce any sensible effect. A hill of an hemispherical figure, three miles high, and six broad, will certainly not, through its attraction, attract the pendulum two minutes out of the true perpendicular : and also it is only in the an excellent bodies of the planets the these pressures are to be perceived, ...

Newton"s idea on the impracticality of such tiny dimensions would revolve out to it is in incorrect. Tiny did Newton know that the scientific revolution that the himself assisted propel would conveniently make together tiny measurements possible.

Weighing the planet using mountains

The very first attempt come "weigh the Earth" was made throughout the French geodesic mission to Peru by Pierre Bouguer, Charles Marie de La Condamine, and also Louis Godin. Their major mission was to recognize the shape of the Earth. Did the planet have an equatorial bulge, together predicted by Newton? (The French had actually sent a various team to Lapland to accomplish the exact same end.) Bouguer provided the expedition as an chance to test Newton"s tip that a mountain would direction a plumb bob from surveyed normal. He determined Chimborazo together the subject mountain. Unfortunately, the measurements came up fully wrong. The plumb bob to be deflected, but in the wrong direction. Bouguer measure a slim deflection far from the mountain (Beeson, webpage).

The following attempt to be the Schiehallion experiment. If surveying the Mason-Dixon line, Charles Mason and also Jeremiah Dixon discovered that periodically their calibrations simply couldn"t it is in made come agree through one another. The cause was that their plumb bobs occasionally deviated native surveyed normal. This discovery led come the Schiehallion experiment performed by Nevil Maskelyne. Unequal Bouguer, Maskelyne did obtain a optimistic result, a deflection that 11.6 arc seconds, and also in the best direction. The observed deflections led Maskelyne to conclude that the mean thickness of the planet is 4.713 times the of water (von Zittel 1914).

It transforms out that Newton"s idea of making use of a hill is essentially flawed. Others tried come repeat these experiments using various other mountains. Numerous measured a negative deflection, as did Bouguer. There"s a great reason because that this. For the same reason that we only see a small component of one iceberg (the bulk is underwater), we only see a small part of a mountain. The bulk of the mountain is inside the Earth. A large isolated mountain should make a plumb bob deviate away from the mountain.

Weighing the planet using small masses

So if using hills is dubious, what does that say about the dubiousness the using small masses that would take months to approach one another even if be separate by mere inches?

This turned the end to it is in a very great idea. Those tiny masses are controllable and their masses can be measured come a high degree of accuracy. There"s no should wait till they collide. Simply measure the force they exert top top one another.

This idea to be the basis for the Cavendish experiment (Cavendish 1798). Cavendish offered two tiny and two huge lead spheres. The two little spheres were hung from opposite ends of a horizontal wooden arm. The wood arm in turn was suspended by a wire. The two big spheres were placed on a separate maker that he could turn to lug a big sphere really close to a small sphere. This nearby separation led to a gravitational force between the small and large spheres, which in turn caused the cable holding the wooden arm come twist. The torsion in the wire acted come counterbalance this gravitational force. Ultimately the system resolved to an equilibrium state. That measured the torsion through observing the angular deviation the the eight from the untwisted state. That calibrated this torsion through a different set of measurements. Finally, through weighing those command spheres Cavendish was able to calculate the mean density of the Earth.

Note the Cavendish did no measure the global gravitational constant G. There is no mention of a gravitational consistent in Cavendish"s paper. The notion that Cavendish measured G is a little bit of historic revisionism. The contemporary notation of Newton"s regulation of universal gravitation, $F=\frac GMmr^2$, merely did no exist in Cavendish"s time. That wasn"t until 75 years after Cavendish"s experiments the Newton"s legislation of universal gravitation was reformulated in terms of the gravitational consistent G. Scientists of Newton"s and Cavendish"s times wrote in regards to proportionalities fairly than utilizing a constant of proportionality.

The very intent the Cavendish"s experiment to be to "weigh" the Earth, and also that is exactly what he did.

Modern Techniques

If the planet was spherical, if there to be no other perturbing impacts such together gravitational acceleration towards the Moon and Sun, and also if Newton"s concept of gravitation to be correct, the duration of a tiny satellite orbiting the earth is offered by Kepler"s third law: $\left( \frac T 2\pi \right)^2 = \frac a^3GM_E$ . Below $T$ is the satellite"s period, $a$ is the satellite"s semi-major axis (orbital radius), $G$ is the universal gravitational constant, and also $M_E$ is the massive of the Earth.

From this, it"s easy settle for the product $G M_E$ if the period $T$ and the orbit radius $a$ space known: $G M_E = \left( \frac 2\pi T \right)^2 a^3$. To calculate the mass of the Earth, all one needs to execute is divide by $G$. There"s a catch, though. If the product is $G M_E$ is known to a high degree of accuracy (and the is), splitting by $G$ will lose a the majority of accuracy since the gravitational constant $G$ is only well-known to four decimal locations of accuracy. This absence of expertise of $G$ inherently plagues any an exact measurement of the mass of the Earth.

I placed a lot of caveats ~ above this calculation:

The earth isn"t spherical. The planet is much better modeled together an oblate spheroid. The equatorial bulge perturbs the orbits that satellites (as do deviations indigenous the oblate spheroid model).The planet isn"t alone in the universe. Gravitation native the Moon and Sun (and the other planets) perturb the orbits of satellites. For this reason does radiation from the Sun and from the Earth.Newton"s concept of gravitation is only approximately correct. Einstein"s concept of basic relativity offers a far better model. Deviations in between Newton"s and Einstein"s theories become observable given an accurate measurements over a long period of time.

These perturbations must be taken right into account, yet the simple idea quiet stands: One have the right to "weigh the Earth" by exactly observing a satellite because that a long duration of time. What"s essential is a satellite specially suitable to the purpose. Below it is:

This is LAGEOS-1, launched in 1976. An similar twin, LAGEOS-2, was deployed in 1992. These room extremely straightforward satellites. They have no sensors, no effectors, no interactions equipment, no electronics. Lock are totally passive satellites. Lock are simply solid brass balls 60 centimeter in diameter, covered with retroreflectors.

Instead, of having the satellite do measurements, world on the ground target lasers at the satellites. That the satellites space covered through retroreflectors means some of the laser light that hits a satellite will be reflected ago to the source. Exactly timing the delay between the emission and also the agree of the reflect light gives a precise measure of the distance to the satellite. Precisely measuring the frequency readjust between the transmitted signal and also the return signal offers a an accurate measure the the price at i m sorry the street is changing.

By accumulating these dimensions over time, researchers can really precisely identify these satellites orbits, and also from that they deserve to "weigh the Earth". The existing estimate of the product $G M_E$ is $G M_E=398600.4418 \pm 0.0009 \ \textkm^3/\texts^2$. (NIMA 2000). That tiny error method this is precise to 8.6 decimal places. Almost all of the error in the massive of the earth is going come come native the apprehension in $G$.

References

M. Beeson, "Bouguer stops working to weigh the Earth" (webpage)

H. Cavendish, "Experiments to recognize the thickness of the Earth," Phil. Trans. R. Soc. London, 88 (1798) 469-526

I. Newton (translated by A. Motte), Principia, The mechanism of the civilization (1846)

NIMA technological Report TR8350.2, "Department the Defense world Geodetic device 1984, Its an interpretation and relationships With regional Geodetic Systems", 3rd Edition, January 2000

K. Von Zittel (translated by M. Ogilvie-Gordon), "History that Geology and also Palæontology to the end of the Nineteenth Century," (1914)