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Newton’s Law of Universal Gravitation states that every object in the universe attracts every other object with a force directed along the line of center-to-center distance between these two objects. The force is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects. This law also maybe written as:

F_{g} = (GMm)/r^{2}

Where F_{g} is the gravitational force, G is the universal gravitational constant

(G = 6.67x10^{-11} N m^{2}/kg^{2}),

M and m are the masses of the objects and r is the center-to-center distance between the two objects.

Newton also noted that the gravitational force of attraction of the sun pulling on an orbiting body must be equal to the centripetal force. (F_{c} = F_{g})

Knowing that F_{c} =ma_{c}, and that centripetal acceleration is equal to

(4π^{2}r)/T^{2},

we can equate F_{c} = F_{g}.

F_{c} = F_{g}

(m4π^{2}r)/T^{2}] = (GMm)/r^{2}

Rearranged, we get

r^{3}/T^{2} = (GM)/(4π^{2})

T is period of the satellite around central object, r is the center-to-center distance from central object.

If carefully noticed, r^{3}/T^{2} is part of Kepler’s Third Law, and is a constant that depends only on the mass of the central object. In this way, Newton was able to provide a theoretical basis for Kepler's laws: orbiting satellites are held in place by gravity.

Collectively, these laws may help us determine a number of things. One such example would be calculating the **mass of a planet**. Clearly it is unrealistic to visit a planet to get its measurements, so we use the measurements we can take from the Earth.

Uranus’ moon Oberon has an orbital period of 13.5 days. Its mean distance from Uranus is 582,600 km. What is the mass of Uranus?

(Note: G=6.67x 10^{-11})

*Solution:*

First find the orbital radius of the moon in meters. (Center to center distance)

582,600 km = 582,600,000 m

Convert the time into seconds.

13.5d x 24h/d x 60m/h x 60s/m = 1,166,400 s

Use the formula: r^{3}/T^{2} = (GM)/(4π^{2})

Rearrange to get mass by itself.

M=(r^{3}4π^{2})/(GT^{2})

Plug in the values.

M=[(582,600,000^{3})4π^{2}]/[(6.67x 10^{-11})(1,166,400^{2})]

M=8.6 x 10^{25} kg

Hint: Don’t be afraid to use parenthesis.

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by Amanda Taphorn

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