# Calculating the Orbital Period

## Homework Statement

Calculate the Orbital Period for the following: Earth's orbit to the Sun. I get stuck towards the end. I must use Newton's second law and have the Universal Law of Gravitation equal the Centrifugal Force.
Newton's gravitational constant: G= 6.67*10^-11 Nm^2/kg^2
Mass of Sun = 1.98*10^30 kg
Mass of the Earth 5.97*10^24 kg
Distance of the Earth from the Sun: 149.6*10^6
T = time

## Homework Equations

Centrifugal Force = (mv^2)/r
v=(2(pi)(r))t
Force of Gravity = (GMm)/d^2
Mm being mass, and G/d^2 being acceleration.

## The Attempt at a Solution

I multiplied Newton's gravitational constant by the mass of the Earth by the mass of the Sun, and then divided it all by the distance of the Earth from the Sun. I got 5.270280882*10^35 km. This was for the gravitational pull.

For the Centrifugal Force, I multiplied the mass of the Earth by v=((2(pi)(r))^2)/t^2. And put it over 149.6*10^6 km. I got (3.52229083*10^34 kg*km)/t^2.

I then set my two results equal to each other. Next, I multiplied by t^2 as that is the variable I am trying to find. This gave me 5.270280882*10^35 km*t^2 = 3.5229083*10^34 kg*km. Then, I divided by km on both sides to cancel it. I now have 5.270280882*10^35 t^2 = 3.5229083*10^34 kg. If I divide by the number on the left side of the equation and then square root both sides, I get a number than can't possibly be the Orbital Period of the Earth.

Can you tell where I am going wrong? Any advice is appreciated.
Thanks.

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mgb_phys
Homework Helper
Centrifugal Force = (mv^2)/r
Force of Gravity = (GMm)/d^2

For an orbit these two are equal so you have
mv^2/r = GMm/r^2

The mass of the Earth cancels, orbit's don't depend on the mass of the small object - that's why a spaceman and a space shuttle can float along in the same orbit.

v^2/r = GM/r^2

Work out v in terms of the circumference and the period ( v= 2 PI r/t) , do a bit of rearranging and you're there.

OK, thanks. I will try this and let you know how it goes.

((4(pi^2)(149.6 km^2))/t^2)/149.6 km=(6.67*10^-11(1.98*10^30 kg))/149.6 km^2

I do not understand what to do. If I cancel the 149.6 km from the denom. on the left side, I can't go much further.